To Understand Post-Counting Numbers and Operations†
Patrick W. Thompson
Luis A. Saldanha
Vanderbilt University
White paper prepared for the National Council of Teachers of Mathematics.
Running head: To understand numbers
†
Preparation of this chapter was supported by National Science Foundation Grant No. REC-9811879. All
opinions expressed are those of the authors and do not reflect official positions of the National Science
Foundation.
Thompson & Saldanha
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Writers of the Standards 2000 have attempted to redress imbalances that became evident after the
first version of the Curriculum and Evaluation Standards [46] and Professional Standards for
Teaching Mathematics [47]. Critics of the original Standards sometimes forget, however, that
they were written amidst a prevailing predisposition in American curricula toward procedures
and computations [44] and a strong predisposition among American teachers toward
calculational discourse [66]. Evidence that students were learning far less than they should was
overwhelming, especially in regard to understanding situations in which mathematics could be
applied [7, 44, 57]. Those first Standards, unfortunately, were interpreted far more literally than
anyone intended. It seems the perceived imbalances were more in readers’ interpretations than in
writers’ intentions.
To some extent, these earlier documents invited the interpretations they got. The
Curriculum and Evaluation Standards often appeared to address pedagogy at the expense of
addressing issues of what should be taught and what students should learn. The Professional
Standards for Teaching Mathematics addressed actions of teaching, but did not address the
substance of teaching — the ideas and methods students should build, and how reformed
instruction might aid their building them.
The Standards 2000 might have a similar fate, but for a different reason. In its first draft,
Standards 2000 writers strove to avoid perceptions of imbalance. Thus, it was replete with calls
for a balance between concepts and skills. But the draft Standards 2000 failed in the same way as
its predecessors to address concepts students should learn and understandings students should
have when they understand those concepts. Thus, though the draft Standards portrayed the
appearance of balance between concepts and skills, it actually gave very little attention to
students’ conceptual understanding. Consequently, attempts to recruit people to a reform
movement that is centered around “student understanding” could have the ironic outcome of
producing a cadre of reformers having incompatible meanings of “to understand” and
incompatible images of what would be acceptable evidence that students understand specific
ideas. We are not saying that that everyone should agree on one view of the understandings
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students should have. Rather, it would be helpful if the Standards were to offer one stance, even
as a sacrificial example.
The importance of explicitness
Writers are often vague about what they have in mind when saying “understand x” or “have a
conceptual understanding of y.”1 Being vague about ideas we want students to learn misses
opportunities to make them explicit, and the community’s ability to debate the value and
intricacies of learning them is hindered severely. As Herscovics [31] noted, many students
experience difficulties learning an idea not because it is inherently difficult, but because they
have developed understandings of ideas it builds on that thwart coherent understandings of the
idea they are trying to learn. By being vague in what we intend that students learn, we fail to
discuss potential obstacles that are due to students having developed understandings other than
ones we intend. We also fail to make explicit the long-term benefits we envision being gained by
students who do develop the understandings we intend.
We appreciate the negative reaction this position engenders in some people. It may seem
anti-postmodern and anti-constructivist to say that we should decide upon understandings
students should have. Our position, however, is that, as professional mathematics educators, we
are remiss not to do so. It is our job to develop thoughtful, informed opinions on what is
important that students understand. Moreover, it is misguided merely to define what we hope
students understand in terms of what teachers should do. That would be anti-constructivist, for it
would amount to saying, “We want students to understand what they are shown.” Put another
way, learning objectives should be stated independently of specific instruction. To mix the two
invites confusion about when a learning objective is achieved.2
1
We use phrases like “concept of x” reluctantly. To say “concept of x” suggests we are comparing a
person’s concepts and a correct view of x. We do not mean this at all. Rather, we mean “conceptual
structures that express themselves in ways people would conventionally associate with what they
understand as x.” But saying that is too cumbersome, so we continue to use “concept of x” and
“understanding of x”.
2
When learning objectives are stated in terms of what teachers should do, then we invite people to judge
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Discussions of understandings we think students should gain from instruction have a
practical side. They focus debate on building consensus on what students should learn from
instruction and how various forms of instruction might support that learning. We note that this
debate is about what students should know and not about skills students should have. The two
debates are related, but they are not identical.
Evidence from many sources points to a lack of substance and lack of coherence in
mathematics instruction in the United States. The TIMMS report of 8th-grade mathematics
instruction in the United States, Germany, and Japan states this clearly.
Finally, as part of the video study, an independent group of U.S.
college mathematics teachers evaluated the quality of
mathematical content in a sample of the video lessons. They based
their judgments on a detailed written description of the content that
was altered for each lesson to disguise the country of origin
(deleting, for example, references to currency). They completed a
number of in-depth analyses, the simplest of which involved
making global judgments of the quality of each lesson’s content on
a three-point scale (Low, Medium, High). (Quality was judged
according to several criteria, including the coherence of the
mathematical concepts across different parts of the lesson, and the
degree to which deductive reasoning was included.) Whereas 39
percent of the Japanese lessons and 28 percent of the German ones
received the highest rating, none of the U.S. lessons received the
highest rating. Eighty-nine percent of U.S. lessons received the
lowest rating, compared with 11 percent of Japanese lessons. [64,
p. iv]
The teachers in the TIMSS video study knew they were going to be videotaped, so it is
reasonable to expect that they “gave it their best.” We also assume that teachers had intentions
about what they hoped students learn, so the preponderance of low ratings suggests that whatever
the U.S. teachers in this study intended, it evidently did not entail coherent mathematical ideas,
students’ generalizations, or student-drawn inferences. That no U. S. lesson’s content received
the highest quality rating and that 89% of the U.S. lessons’ content received the lowest quality
rating points to an incoherence, among this sample’s U.S. teachers, regarding what students
that a learning objective has been achieved once the teacher has done what was prescribed —
regardless of whether students learned what was intended.
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should learn. This makes it even more imperative that mathematics educators engage in open
discussions of the ideas and modes of reasoning we intend students’ schooling to foster. One
major source of personal reform is when teachers realize that what they are teaching does not
support what students should be learning.
We hope no one interprets us as “teacher bashers”. We are mathematics teacher educators
as well as researchers; we work daily with teachers and prospective teachers. We are
mathematics teachers ourselves. However, we also have a “public health” perspective on
problems of mathematics teaching. A community resolves problems most effectively when it
discusses their scope, severity, and sources openly and objectively.
A new discourse about what students should understand
Our concern for inadequate explications of meanings is not merely a matter of rhetorical
preference. Rather, without explication, many readers will trivialize statements about students’
understandings of number and operations or interpret them inappropriately. We feel justified
making this statement on the basis of a number of research studies. We cited one result of the
TIMSS video study, that 89% of U.S. lessons received the lowest quality rating. We doubt that
any of the teachers in this sample would claim that they deliberately chose to teach low-quality
content. Moreover, this result is especially powerful because the TIMSS sampling technique was
to draw nationally-representative samples from each of its participating countries [64].
Two additional studies are Post, Harel, Behr, and Lesh [53] and Ma [43]. Post et al. [53]
gave several versions of a ratio and rate test to 218 intermediate mathematics teachers in Illinois
and Minnesota. The test reflected concepts covered in the intermediate (4-6) mathematics
curriculum. Teachers scored between the13-year-old NAEP average and the17-year-old average
on items drawn from the 1979 NAEP; overall performance among teachers varied widely across
test versions, but average performance scores ranged from 60% to 69% across test versions, and
more than 20% of the teaches scored less than 50%.
An important result in the Post et al. study is teachers’ performance on problems like this:
“Melissa bought 0.46 of a pound of wheat flour for which she paid $0.83. How many pounds of
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flour could she buy for one dollar?” [53, p. 193]. Only 45% of the teachers answered this
question correctly and 28% left the page blank or said “I don’t know.” Teachers were also asked
to explain their solutions as if to a student in their class. In the case of the “Melissa” problem
(above), only 10% of those who answered correctly could give a sensible explanation of their
solution. The authors concluded:
Our results indicate that a multilevel problem exists. The first and
primary one is that fact that many teachers simply do not know
enough mathematics. The second is that only a minority of those
teachers who are able to solve these problems correctly were able
to explain their solutions in a pedagogically acceptable manner.
[53, p. 195]
Ma [43] compared elementary school teachers’ understandings of mathematical topics
they commonly taught. She found that despite the U.S. teachers’ advantage in educational
backgrounds3, Chinese teachers were far more likely than U.S. teachers to exhibit richly
connected and pedagogically powerful understandings of what they expected students to learn.
Thus, both the Post et al. and Ma studies point to the distinct possibility that phrases like “deep
understanding” and “with meaning” will not communicate well without careful, detailed
explication.
Numbers and Operations
Before discussing what it means to understand post-counting numbers and operations, we will
relate our method of conceptual analysis to techniques in mathematics for constructing number
systems formally, and we will discuss integers as a number system from mathematical and
psychological perspectives.
Mathematical foundations and mathematics education
Mathematicians rely heavily on symbol systems to aid their reasoning. Symbol systems are tools
for them. Mathematicians strive to develop symbol systems (inscriptions and conventions for
3
Chinese elementary school teachers enter normal school after ninth grade, graduating two to three
years later.
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using them) that capture essential aspects of their intuitive understandings and means of
operating, so that they need not rely explicitly on conceptual imagery and operations as they
move their reasoning forward or generate further insight. This has been a hallmark of
mathematical activity as long as there has been mathematics.
In the 18th and 19th centuries, mathematicians found that their symbol systems, used
according to established conventions, led to contradictory results. Many mathematicians saw that
the problem was in their intuitions of number, not in the symbol systems they had devised. The
idea of “mathematical development of number systems” arose from the need to make commonly
held meanings for number more precise. This development was influenced also by historical
accounts of the emergence of new numeric understandings, such as eventual acceptance that
“nothing” is a numeric amount, that deficits can be measured, and that number systems could be
extended even if no one understood what they were.4
Number systems from two perspectives
In discussing the development of number systems we will adopt a useful yet completely arbitrary
distinction. It is between “mathematical numbers” and “psychological numbers”. This is not a
factual distinction. Rather, it is a rhetorical ploy that gives greater ease to speaking of numbers
from two points of view. The first is the perspective of mathematicians who work with numbers
at advanced levels; the second is from the perspective of psychologists who are interested in
cognitive processes that are active when people reason numerically. Mathematical numbers are
mathematicians’ attempts to explicate psychological numbers.
By “mathematical numbers” we mean number systems as developed within a method for
expressing structural characteristics of systems that they understand intuitively but imprecisely.
The primary aspect of this method is that of “extension by embedding”. By “psychological
numbers” we mean understandings people have, in mature or formative stages, that correspond
4
As was the case for complex numbers, and later with numbers systems like hyperreal numbers which
are used in non-standard analysis [30, 65], surreal numbers, and quaternions, which have no known use
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roughly to what mathematicians attempted to capture formally.
Mathematically, a number system is a set of objects called together with operations
defined on elements of that system which satisfy certain properties. Numbers do not exist apart
from the operations defined on them.
New number systems are defined in terms of already-defined ones. One mathematical
motivation for extending a number system to a larger sphere is to define a system that is closed
under operations that weren’t closed before. The whole numbers are not closed under subtraction
(3 - 7 is not a whole number), integers are closed under subtraction but not under division (7 ÷ 3
is not an integer), rational numbers are closed under division by non-zero divisors but not under
root extraction ( 3 5 is not a rational number).
Integers
According to our rhetorical ploy of distinguishing between mathematical and psychological
numbers, we will give a brief overview of integers as developed mathematically and a brief
overview of integers as psychological constructs. In doing so we hope to draw a parallel between
them to show that the mathematical development does have psychological implications.
Integers as mathematical numbers
Constructing the integers mathematically from whole numbers means to develop a new number
system which has in it a subsystem identical to the whole numbers, which preserves addition and
subtraction of whole numbers, and which also extends the whole numbers so that it (the new
system) is closed under subtraction. A common technique for extending a number system so that
the extended system is closed under an operation is to create pairs from the original system and
then partition those pairs into classes. The classes are defined so that every pair in a class would
produce the same result under the operation were the old system actually closed. This defines the
elements of the new system.
except to challenge prevailing intuitions of number systems [14].
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For example, even though 2-5 is undefined in the whole numbers, 2-5, 3-6, 4-7, and etc.
all would be the same as 2-5 if it were defined. In each case the subtrahend exceeds the minuend
by 3. Each pair of whole numbers in this list represents the result of subtracting a subtrahend that
is 3 larger than the minuend. Put another way, an integer is determined by a pair of whole
numbers. Because many different pairs have the same difference, whatever an integer is, it can
be determined by an infinite number of whole-number pairs. The new system still needs a formal
criterion for when two whole-number pairs determine the same difference. Moreover, the
criterion cannot use subtraction to define it, because you can’t say “two pairs are equivalent if
they have the same difference (result of subtracting)” when we are not guaranteed that either has
a difference at all.
Figure 1 shows two whole-number pairs — (a,b) and (c,d) — that have the same
difference. Since the two pairs have the same difference, we can “rotate” the bottom pair in
Figure 1 and fit the two pairs together, as in Figure 2. This gives a rule of equivalence — two
pairs of whole numbers (a,b) and (c,d) are equivalent (have the same difference) if and only if
(a+d) = (b+c).
a
b
c
d
Figure 1. Two pairs of whole numbers that have the same difference.
a
b
c
d
Figure 2. (a,b) and (c,d) have the same difference if and only if (a + d) = (b + c).
Since integers are somewhat like differences of whole numbers (not the actual result of
subtracting, but the pair of whole numbers subtracted), adding integers must be like adding
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differences. To continue the present example, if (a,b) and (c,d) are two whole-number pairs that
each represent a difference, then we must define their sum so that this sum represents a
difference that is the sum of (a,b)’s and (c,d)’s differences. The pair (a+c,b+d) represents a
comparison that has as its difference the sum of (a,b)’s and (c,d)’s differences (Figure 3).
a
c
b
d
Figure 3. The pair (a+c,b+d) has the sum of differences in (a,b) and (c,d) as its difference.
We hasten to point out that in Figures 1-3 the pairs (a,b), (c,d) etc. are not integers. The
integer normally represented as “+5” is defined to be the class of whole-number pairs equivalent
to (5,0) — the class of pairs in which the first entry exceeds the second by 5. This is represented
as [(5,0)]. The integer normally represented as “-3” is defined to be the class of whole-number
pairs equivalent to (0,3) — the class of pairs in which the second entry exceeds the first by 3,
which is represented as [(0,3)]. That is, any specific comparison of whole numbers is but one
instance of a class of equivalent comparisons. This is similar to the relationship between a
specific person and the species Homo sapiens. A person is a Homo sapiens, and is equivalent in
species to any other member of Homo sapiens, but Homo sapiens is not a specific person.
This approach has a number of immediate consequences. First, every integer has an
additive opposite. [(a,b)] and [(b,a)] are additive opposites, for the pair (a+b,b+a) has a
difference of 0. So, -[(a,b)], the negative of [(a,b)], is defined to be [(b,a)].Put another way, the
additive opposite of any integer x is the integer that is comprised by the class of pairs having
differences of the same size as those making x, but the comparisons are in the opposite direction.
Another immediate consequence of this approach is that subtraction of integers can now be
defined in terms of addition of integers, for now the concept of “opposite” is well defined.5
5
We strongly criticize textbooks that define integers as “whole numbers together with their opposites.”
The concept of opposite is founded upon the concept of integer, so defining integers in terms of opposites
is tantamount to relying on students’ thoroughly understanding what you intend to define for them.
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Integers as psychological numbers
The prior section provided a brief mathematical development of integers. But it did not address
what people understand as they develop a concept of integers that is consistent with the
mathematical development. Equivalence classes of whole-number pairs are a convenient
definition of integers, but they do not convey what people might have in mind that an
equivalence class formalizes. Explicating that requires a conceptual analysis of integers,
specifically an analysis of what constitutes a difference of two whole numbers.
Psychological integers emerge as people develop general images of quantitative
comparisons and changes [8, 29, 75, 78]. We try to capture this in Figure 4 and Figure 5. Figure
4 shows that, early on, children can conceive specific comparisons in specific settings. Figure 5
shows that, later, children can conceive specific comparisons as being gotten from arbitrary
quantities whose only salient characteristic is that they differ by the amount in question.
Specific comparison
determines a specific
excess
Specific comparison
determines a specific
deficit
Figure 4. Early conceptions of difference.
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Generic comparison
determines a specific
excess
Generic comparison
determines a specific
deficit
Figure 5. Later conceptions of difference.
Vergnaud [78] and Thompson [71, 75] point out the great difficulty students have
reasoning relationally about situations typically identified with integer reasoning. They argue
that it is students who develop the ability to reason about transformations of quantities who also
grasp integers as numbers in themselves. For example,
1. James played two games of marbles. In the first, he won 7 marbles. In
the second, he lost 10. What has happened altogether?
Upon reading Problem 1 primary and middle-school students commonly think that James has 7
marbles after the first game, and therefore he cannot have lost 10 marbles in the second – he
could only have lost 7. That is, they interpret both “7” and “10” as referring to numbers of
marbles that James has at different times, and they are all the marbles he has at those times. This
is entirely natural, and is a productive basis for learning elementary mathematics [8, 9]. To
conceive the entire situation coherently, however, they need to think of “7” and “10” in one of
two ways. One way is that the situation is about changes in James’ numbers of marbles. Another
is that the situation is about excesses or deficit in his number of marbles at one moment in time
when compared with his number of marbles at another moment in time. Either way it becomes
evident that, altogether, James lost 3 marbles. Whatever number of marbles he had before the
first game, he had 3 fewer after the second. The need to have a different conception of numbers
is even more evident in the case of Problem 2, a follow-up to Problem 1.
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2. Is it possible that James ended the first game owing 3 marbles?
In the case of Problem 2, if we think of James’ initial number of marbles as a whole number of
marbles then are unable to understand that he might have started Game 1 in debt. In this case,
numerals do not refer to numbers of marbles. They refer to relationships. To end the first game
having -3 marbles, James must have begun it having -10 marbles. This points out the need for
students to rethink even the basic notion of “having” a certain number of marbles. “Having” -10
marbles makes sense only if we understand “having” as meaning net worth. A net worth of -10
marbles refers to the deficit that results from comparing the number of marbles James possesses
(or could possess if he called in all his debts) with the number of marbles he owes other people.
Similarly, “having” 10 marbles would not mean to possess 10 marbles. Rather, it would mean
that you possess (or could possess if you collect debts owed you) 10 marbles more than you owe.
You could actually possess any number of marbles and have a net worth of 10 marbles.
Figure 6 emphasizes that, psychologically, whole numbers are not integers. Whole
numbers, as psychological numbers, are understood as sizes of collections. Integers, as
psychological numbers, are not sizes of collections. They are amounts of excess, deficit, or
change determined by a comparison of amounts. The integer “+5” does not refer to five things.
Rather, it refers to the result of comparing two quantities of which one has five more than the
other, or to changing one amount into another by adding 5 to it. As such, to understand “+5” as an
integer, one must assimilate it to a scheme of operations that entails the potential transformation
of a class of situations [78]. Put another way, whereas understanding “5” as referring to a number
of things means to assimilate it to a scheme of operations that creates a cardinality,
understanding “+5” as an integer means to assimilate it to a scheme that creates relationships
between pairs of cardinalities. A scheme for integers would allow students to see immediately
that “net worth of $5000” could refer to an infinite number of possibilities — such as a person
having $1,000,000 in assets and $995,000 in liabilities or a person having $6000 in assets and
$1000 in liabilities.
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Whole numbers: Numbers of things
0 1 2 3 4 5 6 7 8 9
… -9 -8 -7 -6 -5 -4 -3 -2 -1 ± 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 …
Integers: Amounts of change or amounts of difference
Figure 6. The whole numbers as “embedded” within the integers.
A scheme for integers is at the heart of understanding how to balance a checkbook. Each
transaction is a transformation of your bank balance, and your bank balance is being tracked at
two locations — at your home and at your bank. Sometimes you have entries in your records that
the bank doesn’t have in its, which means that you have transformed your balance by those
transactions, but the bank has not. You have two avenues to see whether you and the bank agree
on what is your balance. You may transform the bank’s balance as if it had received the items it
hadn’t, by subtracting from its balance the amounts of uncleared checks and by adding the
uncleared deposits. Or, you may “undo” those transactions in your records that are still on their
way to the bank, thus adding to your balance the amounts of uncleared checks and by subtracting
from your balance any uncleared deposits. In either case the underlying idea is of keeping track
of a balance’s transformations, not just of check or deposit amounts.
Dep Hayward 48.32
2015 Billiard Co. 22.41
My Checkbook Register
Bank’s Version of
My Checkbook Register
Num
2012
2013
2014
Dep
Item
Elect Co.
Gas co.
Albertsons
Fullers
Amt
-113.24
-94.23
-193.48
+782.82
Bal
2234.81
2140.58
1947.10
2729.92
Num
2012
2013
2014
Dep
2015
Dep
2016
Item
Elect Co.
Gas co.
Albertsons
Fullers
Billiard Co.
Hayward
Babysitter
Amt
-113.24
-94.23
-193.48
+782.82
-22.41
+48.32
-17.50
Bal
2234.81
2140.58
1947.10
2729.92
2707.51
2755.83
2738.33
2016 Babysitter 17.50
}
Figure 7. A bank balance as recorded in two different registers.
Reasoning relationally is in general more difficult than reasoning about the items related.
Pallermo [50] noted this in his study of children’s difficulties understanding “less” and “fewer”.
Thompson and Dreyfus [75] noted likewise that the 6th-graders in their study repeatedly
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interpreted descriptions of changes in a turtle’s location as if describing the turtle’s ending
location. Vergnaud [78] found in a series of studies that children often confused the states related
by a transformation with the transformation itself.6
As noted in the prior section, rules for processing sign-augmented numerals do not make
operations on integers. Learning objectives regarding operations on integers must be coherent
with learning objectives regarding integers themselves.7 If we expect students to understand
integers as amounts of excess or deficit, or as amounts by which quantities change, then any
learning objectives for operations on integers must take this into account. As such, expressions
like -(-2 - -4), to be sensible to students, cannot be about whole number amounts. They must be
about excesses, deficits, or transformations of amounts. Of course, students can attempt to master
rules for processing sign-augmented numerals without any understanding of what they are about.
But such attempts are short sighted and produce results like those in national evaluations of
mathematical performance and international studies of mathematical performance.8
Multiplicative numbers
The 1996 National Assessment of Educational Progress [54] gave the following items to 8th and
12th graders.
3. Luis mixed 6 ounces of cherry syrup with 53 ounces of water to make a cherryflavored drink. Martin mixed 5 ounces of the same cherry syrup with 42 ounces of
water. Who made the drink with the stronger cherry flavor? Give mathematical
evidence to justify your answer. (NAEP M070401)
6
This is very similar to Piaget’s [51] observation that young children see “being ahead” as “being faster”.
However, one car being ahead of another is a comparison of their locations relative to some common
point, whereas one car being faster than another is a comparison of transformations of their locations in a
given amount of time.
7
To some people integers are no more than marks on paper. We exclude them from this remark,
intending to include those people whose schemes allow them to see equivalence among comparisons of
excess and deficit.
8
The 1996 Nation’s Report Card [54] reported that over 50% of 8th graders in its national sample
answered this question incorrectly: The lowest part of the St. Lawrence River is 294 feet below sea level.
The top of Mt. Jacques Cartier is 1277 feet above sea level. How many feet higher is the top of Mt.
Jacques Cartier than the lowest point of the St. Lawrence river? (NAEP M069301)
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1980 Population
4.
1990 Population
Town A
Town B
= 1000 people
= 1000 people
In 1980 the populations of Towns A and B were 5000 and 6000, respectively. In 1990
the populations of Towns A and B were 8000 and 9000, respectively.
Brian claims that from 1980 to 1990 the two towns’ populations grew by the same
amount. Use mathematics to explain how Brian might have justified his answer.
Darlene claims that from 1980 to 1990 the population of Town A had grown more.
Use mathematics to explain how Darlene might have justified her answer. (NAEP
M069601
Question 3’s intent was to see the extent to which students could quantify intensity of
flavor (higher ratio of cherry juice to water means more intense cherry taste). Question 4’s intent
was to see whether students could compare quantities additively (by difference) as well as
multiplicatively (by ratio). Compared additively, both towns grew the same amount (1000
people). Compared multiplicatively, Town A’s 1990 population was 8/5 (160%) as large as its
1980 population, whereas Town B’s 1990 population was 9/6 (150%) as large as its 1980
population.
While these problems might seem straightforward, they challenged 8th and 12th grade
students who took part in the1996 National Assessment of Educational Progress [54]. Less than
half the 12th graders gave even partially acceptable answers to Problem 3; about one-fifth of 8th
and one-fourth of 12th graders gave partially-correct responses to Problem 4 (Table 1). Though
NAEP students’ performance was poor, it was consistent with a long line of studies that
examined students’ abilities to reason about ratios and relative amounts [25, 26, 28, 33, 34, 48,
49, 76]. Nevertheless, U.S. students have done poorly on such items even in comparison to other
countries [16, 44].
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Table 1. Student performances in 1996 NAEP on Questions 3 and 4 (* data not released)
Grade
8
12
Problem 3
Correct:.......................... *
Partially correct ............. *
Incorrect ........................ *
Omitted.......................... *
Off task.......................... *
Correct:.....................23%
Partially correct ........26%
Incorrect ...................42%
Omitted.......................9%
Off task.......................1%
Problem 4
Correct: ......................... 1%
Partially correct........... 21%
Incorrect...................... 60%
Omitted ....................... 16%
Off task ......................... 2%
Correct: ......................... 3%
Partially correct........... 24%
Incorrect...................... 56%
Omitted ....................... 16%
Off task ......................... 1%
Explanations of US students’ poor performance on questions like these are complicated.
One reason is that the understandings tapped by these questions can “go wrong” at many
developmental junctures during students’ schooling. As such, we will focus on describing one
path to successful development of schemes that would support success on these and even more
sophisticated problems.
We will continue the approach used for integers by discussing rational and irrational
numbers from mathematical and psychological perspectives. The mathematical treatment will be
quite brief. A full treatment is beyond the scope of this chapter.
Fractions and more as mathematical numbers (rational and real numbers)
The mathematical construction of rational numbers is tremendously general and abstract for the
same reasons that the mathematical definition of function is general and abstract. The
construction in its present form addresses a history of paradoxes and contradictions, and the
present definition is the result of a long line of accommodations to eliminate them. For example,
one motivation for the modern development of rational numbers is a spin-off from the period in
which the calculus was arithmetized (grounded in the analysis of real-valued functions; see [18,
83]). The original notion of derivative as a ratio of differentials (first-order changes in two
quantities’ values) fit with the notion of rate of change as a relationship between two varying
quantities. But contradictions that traced back to notions of derivative-as-ratio led d’Lambert, for
one, to wonder whether “ dy dx ” should be thought of as merely a symbol that represents one
number instead of as a pair of symbols representing a ratio of two numbers [17]. Questions about
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rates-as-numbers and about continuity of functions threaded themselves into formal
constructions of rational and real number systems. Rational numbers were constructed as
equivalence classes of integer pairs — a rational number being the thing represented by any of a
class of equivalent fractions. Real numbers were constructed as above-bounded classes of
rational numbers (by Dedikind’s approach) or as equivalence classes of convergent sequences of
rational numbers (by Cauchy’s approach). The method of extension from the integers to a system
closed under non-zero division creates the rational number system. The method of extension
from the rationals to a system closed under convergence of rational-number sequences creates
the real number system.9
We hasten to add two points. First, the mathematical developments of rational and real
number systems interconnect so many issues that they are typically not treated until advanced
undergraduate or introductory graduate mathematics courses. As such, we have absolutely no
idea what school mathematics textbook authors or other writers intend when they say they want
middle-school students to “understand” rational and irrational numbers. To us, understanding
either number system, as a system, means to understand how it is created, why it was created and
what problems were solved by its creation, and the issues are hardly within reach of very bright
high school students after studying pre-calculus functional analysis. So, to us, using the terms
“rational numbers” and “real numbers” with school students introduces a false rigor and cannot
be productive. What are they supposed to understand about numbers when they see a
demonstration that
2 cannot be represented as a fraction? This does not prove that
irrational, for it is not an existence proof. It doe not prove that
2 is
2 actually exists. Rather, it
proves that there is no rational number that, squared, gives 2. Even if it were an existence proof
(as in, “all roots exist and any root that is not rational is irrational”), the logic of this proof
9
The reason that “convergence of sequences” is an important criterion for closure is that there needs to
be a number to approximate that corresponds to what we think we are approximating with ever more
precise rational approximations. That is, we are assured, within this new system, that there is a square
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implies that complex numbers are irrational. Even more, in order to say that the standard proof of
2 ’s non-rationality proves that
2 is irrational, we must presume that students understand the
real numbers (as rational and non-rational real numbers) in order to introduce irrational numbers
as a subset of them. That is, we must presume they already understand what we intend they
learn.10
Second, the fact that mathematical developments of rational numbers and real numbers
interconnect a myriad of issues at the foundations of mathematics makes a direct mapping
between these number systems and psychological numbers impossible. We cannot look to the
mathematical development of rational and real numbers for guidance on how to think about
students’ thinking and its development. Rather, the mathematical development of these number
systems provides strong background constraints on how we envision the end products of
successful conceptual development.
Fractions and more as psychological numbers
Our discussion of psychological numbers in this section will take a somewhat circuitous route
from examples of mildly sophisticated reasoning, to discussing factors in its development, to a
discussion of more general characteristics of advanced multiplicative reasoning and some
relationships with mathematical rational and real numbers. In our discussion we will draw
heavily from foundational research by Gerard Vergnaud [79-81], Les Steffe and Ron Tzur [59,
62, 77], Tom Kieren [35, 36, 38], the Rational Number Project [2, 3, 40, 41, 52, 53], Jere
Confrey [11, 12] and our own work in quantitative reasoning [55, 69, 71-74]
As noted already, Post et al. [53, p. 193] gave this problem to a sample of intermediateschool mathematics teachers: Melissa bought 0.46 of a pound of wheat flour for which she paid
root of 2, even if we know we cannot calculate it exactly.
10
We suspect that most students will think that irrational numbers are relatively rare, since they know of
so few. There are infinitely many irrational numbers, and the order of infinity is larger than that of the
rationals. One way to imagine the vast difference in numbers is this: If stars in the universe represent
rational numbers, then the irrational numbers fill the empty space between them.
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$0.83. How many pounds of flour could she buy for one dollar? A standard solution would be to
set up an equation, as in
.46 x
= , and then solve for x. However, as Post et al. found, setting up
.83 1
this equation correctly and understanding what it represents are not the same. A solution to this
problem that relies on reasoning instead of on equations might go like this: If .46 lb. costs $.83,
then 1/83rd of .46 lb. costs $.01. Thus 100/83 of .46 lb. costs $1.00. So, about 5/4 of.44 lb. costs
$1.00, or about .55 lb. costs $1.00.11 A more sophisticated expression of the same reasoning
would be $1.00 is 100/83 as large as $.83, so you can buy 100/83 of .46 lb. for $1.00. The latter
reasoning, is, in effect, what one would do when solving the equation
.46 x
= by multiplying
.83 1
each side by 1 and dividing both sides by .83.
What conceptual development leads to such reasoning? A variety of sources suggest it is
through the development of a scheme of conceptualizations of measurement, multiplication, and
division, conceptualizations that build on each other and complement one another. Note that we
speak of conceptualizations of measurement, fractions, multiplication, and division. This is not
the same as measuring, multiplying, and dividing.
Measurement schemes
To conceive of a measured quantity is to imagine the measured attribute as segmented [45, 61] or
in terms of a coordination of segmented quantities [51, 56, 72]. The idea of ratio is at the heart of
measurement. To conceive of an object as measured means to conceive of some attribute of it as
segmented, and that segmentation is in comparison to some standard amount of that attribute. A
mile, measured in yards, is 1760 yards because the length of one mile is 1760 times as long as
the length of one yard. That same distance is 5280 feet because it is 5280 times as long as the
length of one foot. It does not mean just that a mile contains 5280 feet, in the sense that a fielded
11
Of course, this is mathematically equivalent to x = 46 ÷ 83, but if students think the point is to go
straight to an equation, they probably will not develop the reasoning that will express itself in writing such
equations from a basis of understanding what is going on.
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soccer team contains 11 players. The distinction is between an additive, part-whole relationship
between a set and its elements and a multiplicative comparison in which the unit is held apart
from the thing measured.
The ratio nature of measurement is trivial to people who have a quantitative scheme of
measurement, but it is nontrivial to children who are building one. A conceptualized measure
entails an image of a ratio relationship (A is some number of times as large as B) that is invariant
across changes in measurement units. For example, Figure 8 illustrates that if m is the measure of
quantity B in units of quantity A (i.e., B is m times as large as A), then nm is the measure of
quantity B in units of 1/nth of A. Conversely, if q is the measure of quantity B in units of
quantity A, then q/n is the measure of B in units of nA. Put another way, the measure of a
quantity is m times as large when you dilate the unit by a factor of 1/m, and its measure is 1/mth
as large when you dilate the unit by a factor of m.
B
B
A
A
Figure 8. Change of measurement unit entails a constant ratio.
A conceptual breakthrough behind the example of Figure 8 is that the quantity of interest
and its comparison with the original unit’s magnitude do not change even with a substitution of
unit.12 Wilki [84] emphasizes this point by distinguishing between numerical equations and
quantity equations. A numerical equation, like W=fd, says how to calculate a particular quantity’s
measure and the result depend on the particular units used. Quantity equations explicate a
12
The first author observed a 4th-grade lesson on the metric system in which children measured their
heights. He asked one boy who had measured his height with a yard stick what he got. “140 centimeters.”
How tall are you? “Four feet seven.” Do you know how many centimeters make 4 feet 7 inches? (Pause.)
“No.”
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quantity’s construction. The equation [W] = [f][d] says that accomplished work, as a quantity, is
created by applying a force through some distance, regardless of your measurement units, and
that the magnitudes remain invariant regardless of unit substitution.
We saw this distinction, between numerical and quantitative formulas, in a 5th-grade
teaching experiment on area and volume. The first author presented the question in Figure 9
during a follow-up interview. Portions of two children’s interviews follow the diagram.
17 in
2
Figure 9. What is the volume of this box?
PT: (Discusses with BJ how the diagram represents a hollow box and what each number
in the diagram represents about it.)
BJ: I don’t know. There’s not enough information.
PT: What information do you need?
BJ: I need to know how long the other sides are.
PT: What would you do if you knew those numbers?
BJ: Multiply them.
PT: Any idea what you would get when you multiply them?
BJ: No. It would depend on the numbers.
PT: Does 17 have anything to do with these numbers?
BJ: No. It’s just the area of that face.
…
PT: (Discusses with JA how the diagram represents a hollow box and what each number
in the diagram represents about it.)
JA: Oh. Somebody’s already done part of it for us.
PT: What do you mean?
JA: All we have to do now is multiply 17 and 6.
PT: Some children think that you have to know the other two dimensions before you
can answer this question. Do you need to know them.
JA: No, not really.
PT: What would you do if you knew them?
JA: I’d just multiply them.
PT: What would you get when you multiplied them?
JA: 17.
To BJ, the formula V=LWD was a numerical formula. It told him what to do with
numbers once he had them. However, it had no relation to evaluating quantities’ magnitudes. To
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the second child, JA, the formula V=LWD was a quantity formula. To him, it was V = (LW)D,
where (LW) produced a measure of a face’s area, and (LW)D produced a volume. JA recognized
that being provided one face’s area was as if “somebody’s done part of it for us,” that part being
the quantification of one face’s area.
Proportional reasoning is important in conceptualizing measured quantities. Vergnaud
[79, 80] emphasized this when he placed single and multiple proportions at the foundation of
what he called the multiplicative conceptual field.13 A single proportion is a relationship between
two quantities such that if you increase the size of one by a factor a, then the other’s measure
must increases by the same factor in order to maintain the relationship. More formally, if x is the
measure of one quantity, and if f (x) is the measure of the other, then f (ax) = af ( x). A double
proportion is a relationship among three quantities, one thought of as a function of the other two,
in which the dependent quantity is proportional to each of the others. More formally, if x and y
are measures of two quantities, and if f(x,y) is a third quantity’s measure, then:
f (ax,by) = af (x,by)
= bf (ax, y)
= abf (x,y).
The concept of torque concerns measuring the twisting force produced by applying a
force at some distance from a resistant hinge or fulcrum. It provides one example of how being
able to conceptualize a double proportion provides a way to think of quantifying a new quantity.
Hinge, or fulcrum
Figure 10. Torque as amount of twisting force
Torque, in its simplest form, can be thought of as “amount of twisting force”, as in “twist
13
Confrey [11] takes a counter position, claiming that multiplicative reasoning emanates from a primitive
operation called splitting, which does not emanate from proportionality but instead underlies it. We cannot
here discuss the two in contrast, as that would take us astray of our present purpose. We will say only
that we agree with Steffe [63] that when we consider the developmental origins of splitting we see that
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my arm.” It is well known that physics students have difficulty understanding its quantification.
However, as a multiplicative concept its quantification can be thought of as this:
(T1) Suppose that applying a force of y force-units (f-units) at a distance of x distance
units (d-units) from a fulcrum produces T twist units.
(T2) Increasing the applied force by a factor of a while applying it at the same distance
increases the twisting force by a factor of a. That is, applying a force of ay f-units at a distance of
x d-units produces a twisting force that measures aT twist units (Figure 11).
Hinge, or fulcrum
Hinge, or fulcrum
x d-units
x d-units
Figure 11. Apply a force a times as great at the same distance produces aT twist units
(T3) Increasing the distance at which the original force is applied by a factor of b
increases the amount of twist by a factor of b. That is, applying a force of y f-units at a distance
of bx d-units produces a twisting force of measure bT twist units (Figure 12).
Hinge, or fulcrum
Hinge, or fulcrum
x d-units
bx d-units
Figure 12. Apply a force at b times the distance from the hinge produces bT twist units.
(T4) Combining (T2) and (T3), we get that applying a force of ay f-units at a distance of
bx d-units produces a twisting force of measure abT twist units.14 If we adopt the convention that
a force of 1 f-unit applied at a distance of 1 d-unit produces 1 twist unit, then a force of u ⋅1 funits applied at a distance of v ⋅1 d-units produces u ⋅ v twist units. This is the conceptual
Confrey and Vergnaud are not in opposition.
14
We stress once again that it is this step, combining T2 and T3, that is enabled by the ability to conceive
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foundation for quantifying torque by the quantity formula “force times distance”.
Multiplication schemes
Contrary to most textbooks, and contrary to Fishbein’s [19] well-known treatise, multiplication is
not the same as repeated addition. To be fair, we should say conceptualized multiplication is not
the same as repeated addition.
The conceptual foundation of multiplication of whole numbers is
quite like the Biblical “multitudes” – creating many from one. That
is to say, multiplication of whole number is the systematic creation
of units of units. At an intuitive level this is repeated addition; at an
operational level it could be either a transformation brought about
by composing [the mental operation of extending one number by
another] with itself a number of times …, or, equivalently, as a
hierarchy of integrations. [68, p. 316]
The difference between conceptualized multiplication and repeated addition is between
envisioning the result of having multiplied and determining that result’s value. Envisioning the
result of having multiplied is to anticipate a multiplicity. One may engage in repeated addition to
evaluate the result of multiplying, but envisioning adding some amount repeatedly cannot
support conceptualizations of multiplication.
Suppose we have three quantities, one of whose measure we think of as being a function
of the other two. By convention the standard unit of the third, dependent, quantity is defined as
that amount made by one unit of the independent quantities. That is, suppose x and y are the
independent quantities’ measures, and suppose f(x,y) is the third quantity’s measure expressed as
a function of the of the two. Then the third quantity’s standard unit is
meas(StandardUnit) = f (1,1)
= 1.
This says that if a double proportion relates the three quantities, then the third quantity’s measure
will be the product of the two other’s measures. We can express this formally as:
a double proportion.
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f (x, y) = f (x ⋅1, y ⋅1)
= x ⋅ y ⋅ f (1,1)
= x ⋅ y ⋅meas(StandardUnit)
= x ⋅ y ⋅1
= x ⋅ y.
The very first step — the understanding reflected in the statement that
f(x,y) = f (x ⋅1,y⋅1) — is the key to moving from multiple proportion to multiplication as the
operation to evaluate a quantity’s measure. This step reflects the cognitive move from thinking
of the values of x and y as numbers of things to thinking them as entailing a ratio comparison
between two quantities’ measures.15 x inches is a number of inches. (x ⋅1) inches specifies that
the thing having measure x inches is x times as large as the same kind of thing having measure 1
inch. This highlights the move from thinking of a length’s measure as a collection of inches to
thinking of its measure as a ratio comparison between the attribute in question and a standard
length (Figure 13.).
12 little lengths, called “inches”
versus
The total length of this
The length of this
Figure 13. Measure as a number of things vs. measure as a ratio comparison. 16
15
A ratio comparison between two quantities is to compare them in terms of relative size — the size of
one is so many times as big as the size of the other.
16
The difference between 12 inches as a collection of inches and 12 inches as a measure that is 12 times
as big as one inch is the difference between seeing a collection additively and seeing it multiplicatively.
Seen additively, comparing 1 inch to the collection entails removing that inch and noticing what is left —
11 inches. Seen multiplicatively, comparing 1 inch to the collection entails removing a copy of 1 inch. The
collection still contains as many inches as before, and yet you’ve pulled out one of its members to
compare against the group.
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This last observation, that understanding objects as entailing a single or multiple
proportion relies on understanding measures as reflecting a ratio comparison, also highlights the
special way in which multiplication must be conceived -- as entailing a multiple proportion.
Generally, most students’ understanding of multiplication does not support proportionality. As
directed by a great deal of curriculum and teaching, they understand multiplication as a process
of repeated addition. But an extensive research literature has documented how quickly this
conception becomes limiting and problematic [15, 19, 24, 27, 42].
Fischbein [19] and Burns (get ref) propose repeated addition as the fundamental initial
idea of multiplication. Because of children’s prior numerical and quantitative experience (which
is fundamentally additive), this conceptualization makes multiplication easy to teach and (in this
limited calculational sense) easy to learn. But thinking about multiplication as repeated addition
leads to severe difficulties in later grades. Specifically, if 5 × 4 means 4+4+4+4+4 -- “add 4 five
times”, then it is difficult to say what 5 32 × 4 means. It can’t mean “add 4 five and two-thirds
times”.
But we need not start where Fischbein, Burns, and most textbooks start. If we start with
the basic meaning of 5 × 4 as “five fours”, then 5 32 × 4 means “(five and two-thirds) fours, or
five fours and 2/3 of four”. The principle difference between “add 4 five times” and “five fours”
is that the former tells us a calculation to perform while the latter suggests something to
imagine.17 In exactly the same way, when we read 5 32 × 4 35 we can understand that we are being
Another way to distinguish between the two is to characterize the nature of question being answered
by the comparison. A collection seen additively entails questions about how much more or less this
collection has in comparison to another. A collection seen multiplicatively entails questions about the
measure of the collection in units of a subcollection.
17
This example carries a larger significance. It is that how students read mathematical statements (i.e.,
the meanings attached to their verbal decoding) can have a large impact on the connections they can
eventually form. If they read “5 times 4” as a command to calculate, then “5 times x” is highly problematic.
But if they read “5 times 4” as “five fours”, this supports the image that we are speaking of a product — (5
times 4) is a number that is 5 times as large as four. “5 times 4” becomes a noun phrase — it points to
something. Thus, understanding “5 times x” as “a number, and it is a number that is 5 times as large as
the number x” can be a natural extension of their understanding of numerical multiplication, if only their
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asked to think about some number of (four and three-fifths). We are being asked to think of 5
(four and three-fifths) and 2/3 of (four and three-fifths).
ADD A PARAGRAPH OR TWO TO DO TWO THINGS: CONNECT THIS IDEA OF
“SOME NUMBER OF SOME THINGS” TO STEFFE’S INTERIORIZED NUMBER
SCHEMES, AND THEREBY SEPARATE THIS FROM CONFREY’S NOTION OF
SPLITTING. SECOND, TALK ABOUT THE ROLE OF IMAGERY AND OPERATIONS,
THEIR DEVELOPMENT, AND THE FACT THAT THIS CAPABILITY EMERGES
THROUGH CHILDREN’S REFLECTION ON ITERATIVE ACTIVITY.
It is important to understand that when we start with the idea that the expression “__×__”
means “some number of (or fractions of) some amount,” we are not starting with the idea that
“times” means to calculate. We are starting with the idea that “times” means to envision
something in a particular way -- to think of copies (or parts of copies) of some amount. This is
not to suggest that multiplication should not be about calculating. Rather, calculating is just one
thing one might do when thinking of a product. Non-calculational ways to think of products will
be important in comprehending situations in which multiplicative calculations might be useful.
The comprehension will enable students to decide on appropriate calculations.
When people write “5 × 4 = __ ” they are suggesting that five fours make some number
of ones, and by convention others understand them as a request to fill in the blank with an
appropriate number. They are asking, “How many ones do we have if we have 5 fours?” The
question, “How much does that make in all?” requests a calculation or some other determination
of total amount.
Though we will discuss fraction schemes in a later section, it is worthwhile to point out
here that to understand multiplication multiplicatively is also to understand fractions as entailing
a proportion. The phrase “quantity X is 1/n of a quantity Y” means that Y is n times as large as X.
understanding of numerical multiplication is appropriately general in the first place.
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It does not mean merely that X is one out of n parts of Y.18 Thinking of 1/n as “one out of n parts”
is to think of fractions additively -- that Y is cut into two parts, one being X and the other being
the rest. It is merely a way to indicate one part of a collection. When students’ encompassing
image of fractions is “so many out of so many”, it possesses a sense of inclusion — that the first
“so many” must be included in the other “so many”. As a result, they will not accept the idea that
we can speak of one quantity’s size as being a fraction of another’s size when they have nothing
physically in common. They will accept “The number of boys is what fraction of the number of
children?”, but they will be puzzled by “The number of boys is what fraction of the number of
girls?”
To think of multiplication as producing some number of some amount, and to think at the
same time of the product (what you make by taking some number of some amount) in relation to
the individual factors, entails proportional reasoning. To understand 5 × 4 multiplicatively, as 5
fours, children need to understand that 4 in 5 × 4 is not just 4 ones, as in 20 = 4 + 9 + 7. Rather,
this 4 is special — it is 1/5 of the product.19 In general, when children understand multiplication
mulitiplicatively, they understand the product (nm) being in a special relationship to n and to m:
m is 1/n as large as (nm) and n is 1/m as large as (nm).
The use of “ugly numbers” sometimes helps clarify the idea that multiplication entails a
18
One could object that nothing is gained by defining other people’s meanings out of existence, even if
they are somehow flawed. This poses an interesting question — to what extent do we want to respect
meanings that don't work and end up hurting children if a particular meaning becomes what they *really*
mean. We understand that a person can say "one out of n,” really mean it additively at the moment of
saying it (like, one of those six boys has a blue shirt), and yet flip effortlessly and unawarely to "the
number of boys over there is six times as large as the number of boys with blue shirts". Our objection is
not to the use of the words "one out of n". Our objection is that mathematics education’s past mistake was
to think that the former automatically conveys the latter. What we want to say is that it takes an incredible,
systematic, programmatic effort over multiple years to ensure that the majority of children internalize the
multiplicative point of view, so that it becomes "the way they see things." It is so important that we are
willing to say, "Blind adherence to the phrase 'one out of n' as conveying the meaning of "1/n" allows
incredible problems to build undetected. It is so problematic that we need to mount a campaign
advocating that teachers and mathematics educators express themselves consistently so that there is no
question what they mean."
19
We are indebted to Jose Cortina for this observation.
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proportional relationship: In 4 73 × 7 23 (read as [four and three-sevenths] seven-and-two-thirds),
1
1
of the product. Now, what does “ 3 “ mean? Recall what we said it
3
47
47
7 23 is special. It is
means for a to be 1/n of b. The statement “a is one-nth of b” means that b is n times as large as a.
Let a=7 23 , b= (4 73 × 7 23 ) , and n=4 73 . Then, to say that 7 23 is
(4 73 × 7 23 ) is 4 73
1
of (4 73 × 7 23 ) means that
4 73
times as large as 7 23 (see Figure 14).
7 23
7 23
7 23
7 23
7 23
Figure 14. (Four and three-sevenths) seven and two-thirds.
One shaded region ( 7 32 ) is
1
of the total, because the total is 4 73 times as large as 7 32 .20
3
47
Division schemes
A long list of research studies point to students’ difficulties in conceptualizing division and its
relationships with multiplication and fractions. Greer [22, 23] and Harel [27] noted the great
frequency with which students will decide upon the use of multiplication or division depending
on the size or type of numbers involved, not on the underlying situation. Graeber [20] found the
same to be true among pre-service elementary teachers. Ball [1] discovered that even college
mathematics majors had difficulty proposing situations in which it would be appropriate to
divide by a fraction. Simon [58] found that pre-service elementary teachers’ knowledge of
division was tightly bound up with their whole-number schemes, meaning, which resulted in
weak connections among various meanings of division and in their inability to identify the unit
of the result. In short, it is unquestionable that division is a problematic area of mathematics
teaching and learning.
20
Of course, the argument is symmetric in regard to both numbers. But to understand the symmetry
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The fact that division is a problematic area does not clarify what is problematic or what
constitutes an understanding of it. In addressing this issue we will begin with a widely used
distinction between two “types” of division settings (sharing and partitioning), and demonstrate
that , among other things, operational understanding of division entails a conceptual
isomorphism between them.
Sharing is the action of distributing an amount of something among a number of
recipients so that each recipient receives the same amount.21 Partitioning is the action of cutting
up an amount in units of some standard amount. “13 people will share 39 pizza’s. How many
pizza’s will each person receive?” usually is classified as a sharing situation, because of the
request to distribute the pizzas into a pre-established number of groups. “You are to give 3 pizzas
to each person, and you have 39 pizzas. How many people can you serve?” usually is classified
as a partitioning situation, because of the request to partition the number of pizzas into preestablished group sizes. It is well known that students often see these situations as being very
different [4, 5, 22, 24], yet both can be resolved by the numerical operation of division.22
How is it possible that the results of sharing and partitioning are evaluated by the same
numerical operation? To most students (and many teachers) this is a mystery, or the issue is not
raised at all. To see why the numerical operation is the same in either situation entails the
development of operative imagery — the ability to envision the result of acting prior to acting —
and to suppress attention to the process by which one obtains those results.
Suppose we have some number of candy bars. Consider this as one mass of chocolate.
•
The process of sharing ends up with each recipient having the same amount of chocolate. So,
presumes an understanding of commutativity.
21
We will speak as if all sharing situations are equi-sharing, and that all partitioning situations are equipartitioning.
22
It is also well known, by these same studies, that if you change the numbers so that in either situation
each person receives 2/3 pizza, students will see these problems even differently yet. We agree with
Confrey [11] that this result is entirely an artifact of persistent experience with stereotypical problems in
which products are always larger than either factor and quotients are always smaller than either minuend
or divisor.
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if we put the chocolate into 7 pieces, each piece contains 1/7 of the chocolate. So, the number
of candy bars in each piece is 1/7 as large as the number of candy bars.
•
The process of partitioning cuts up the mass of chocolate into a number of pieces of a given
size (perhaps with an additional piece that is a fraction of the given size). Suppose we cut up
this mass into pieces, each piece the size of 7 candy bars, and consider any remaining bars to
make some fraction of a piece (i.e., of 7 candy bars). We can determine the number of pieces
by removing one candy bar from each piece and a proportionate part of a candy bar from any
fraction of a piece. Each piece (or fraction of a piece) is 7 times as large as what we remove
from it, so the total is 7 times as large as what we remove in all. The mass of chocolate is 7
times as large as what we remove, what we removed counts the number of pieces, therefore
the number of pieces is 1/7 as large as the number of candy bars.
The two analyses above show that, in either case, the result of sharing or the result of
partitioning produces a number that is 1/7 as large as the number of candy bars. The above
analysis also reveals that fractional reasoning and proportional reasoning are integral to
understanding that partitioning and sharing are equivalent.
Finally, students can see that measuring and partitioning, and therefore measurement and
division, are equivalent even though at the level of activity they appear to be very different.
When they understand the numerical equivalence of measuring and partitioning they understand
that any measure of a quantity induces a partition of it and that any partition of a quantity induces
a measure of it.
Fraction schemes
A common approach to teaching fractions is to have students consider a collection of objects,
some of which are distinct from the rest, as depicted in Figure 15. But, what might Figure 15
illustrate to students? Three circles out of five? If so, they see a part and a whole, but not a
fraction. Three-fifths of one? Perhaps. But depending on how they think of the circles and
collections, they could also see three-fifths of five, five-thirds of one, five-thirds of three, or fivethirds of three-fifths (see Figure 16).
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Figure 15. What does this collection illustrate?
They could also see Figure 15 as illustrating that 1 ÷ 3 5 = 12 3 —that within one whole
there is one three-fifths and two-thirds of another three-fifths, or that 5 ÷ 3 = 1 2 3 —that within 5
is one 3 and two-thirds of another 3. Finally, they could see Figure 15 as illustrating
5
3
× 3 5 = 1—that five-thirds of (three-fifths of 1) is 1.
If we see
as one collection, then
is one-fifth of one, so
is three-fifths of
one.
If we see
as one collection, then
is
one-third of one, so
is five-thirds
of one.
If we see as one circle, then
circles, so
is one-fifth of five, and
three-fifths of five.
is five
is
If we see as one circle, then
is three
circles, so
is one-third of three, and
is five-thirds of three.
Figure 16. Various ways to think about the circles and collections in Figure 15.
Figure 15 is customarily offered by texts and by teachers to illustrate 3 5 as three out of
five. We rarely find texts or teachers discussing the difference between thinking of 3 5 as “three
out of five” and thinking of it as “three one-fifths.” How a student understands Figure 15 in
relation to the fraction 3 5 can have tremendous consequences. When students think of fractions
as “so many out of so many” they are puzzled by fractions like 6 5 . How do you take six things
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out of five?23
Kieren [35, 37, 38] and the Rational Number Project [2, 3, 40] have given the most
extensive analyses of rational number meanings. Their approach was to break the concept of
rational number into subconstructs,—part-whole, quotient, ratio number, operator, and
measure—and then describe rational number as an integration of those subconstructs. Our
feeling is that their attempt to map their systems of complementary meanings into the formal
mathematical system of rational numbers will necessarily fall short. As we already noted, many
of the mathematical motivations for developing the rational numbers as a mathematical system
did not emerge from meanings of subconstructs. Rather, it emerged from the larger endeavor of
arithmetizing the calculus (see prior discussion). Moreover, our analysis places the idea of
psychological rational numbers squarely within multiplicative reasoning as a core set of
conceptual operations, and prior analyses were less focused on conceptual operations per se.
A fraction scheme is based on conceiving a reciprocal relationship between two
quantities in terms of relative size. Amount A is 1/n the size of amount B means that amount B is
n times as large as amount A. Amount A being n times as large as amount B means that amount
B is 1/n the size of amount A.
We worded these sentences very carefully, avoiding saying “1/n of”. This is to emphasize
the slippery connection, already mentioned earlier, between comparison and inclusion. Fraction
sophisticates can say “A is some fraction of B” and get away with it because they imagine that A
and B are separate amounts (see Figure 13). Children are often instructed, and therefore learn,
that the fractional part is contained within the whole, so “A is some fraction of B” connotes a
sense of inclusion to them, that A is a subset of B. As a result statements like “A is 6/5 of B”
makes no sense to them.
23
We cannot count the number of times teachers, in-service and pre-service, have said “change
and they’ll understand.” This misses the point that it is problematic if we must change 65 it to
means that students cannot understand fractions as entailing a proportional relationship.
DRAFT - 33
6
5
to
1 51 , for it
1 51
Thompson & Saldanha
To understand numbers
To say “A is m n as large as B” means that A is m times as large as 1 n of B. So, “A is
6/5 as large as B” does not mean that there are six of something in A. Rather, it means that A is 6
times as large as (1/5 of B).
Our conceptual analysis of fraction knowledge resembles the approach taken by Lesh,
Behr, Post, and Harel [2, 3, 41]. They characterized m/n of B as m one-nths of B. The difference
between their approach and ours is that we point out the need for a direct link to students’
conceptualizing reciprocal relationships of relative size. Without our insistence on students
having that link they could appear to be speaking of fractions in ways we intend while in fact be
thinking of additive inclusion, as we described earlier.
Students gain considerable mathematical power by coming to understand fractions
through a scheme of operations that express themselves in reciprocal relationships of relative
size. For example, “related rate” problems are notoriously difficult for U.S. algebra students.
It takes Jack 20 seconds to run as far as Bill does in 15 seconds.
Bill’s top speed is 18 ft/sec. About how fast does Jack run?
One way to reason through this using basic knowledge of fractions is as below. The chain
of reasoning is based on Figure 17. In it we see that Jack and Bill ran the same distance but in
different amounts of time. It is important to note that 5 seconds is 1/4 of Jack’s time and 1/3 of
Bill’s time.
20 secs
15 secs
1/4 of Jack’s time
1/3 of Bill’s time
Figure 17. Jack and Bill took different amounts of time to run the same distance.
•
Bill runs 1/3 of the distance in 1/3 of his time.
•
Jack runs 1/4 of the distance in 1/4 of his time.
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•
1/3 of Bill’s time is as large as 1/4 of Jack’s time.
•
If Bill were to continue running another 1/4 of Jack’s time, he and Jack will have run
for the same amount of time.
•
If Bill were to continue running another 1/4 of Jack’s time, he would run an
additional 1/3 of the distance
•
Jack runs 3/4 as far as Bill in the same amount of time
•
Therefore, Jack runs 3/4 as fast as Bill. So he runs at about 13.5 ft/sec.24
Fractions as reciprocal relationships of relative size also draw on relationships among
measure, multiplication, and division. For example, suppose we interpret “a ÷ b” as “the number
of b’s in a”. Then whatever number is a ÷ m n (i.e., whatever number is the number of m n in a)
it is n times as large as a ÷ m, because m n is 1 n as large as m. (see the discussion following
Figure 8). When we recall that a÷m is 1 m as large as a (see discussion of equivalence between
partitioning and sharing), we can conclude that whatever number is a ÷ m n , it is n times as large
as 1 m of a. Put more briefly, a ÷ m n = n m ⋅ a . “The number of m n ’s in a is n times a large as
1
m of a.” This is a conceptual derivation of the “invert and multiply” rule for division by
fractions, and its interpretation is straightforward when thinking in terms of relative sizes.
Similarly, we can use fraction relationships to reason about algebraic statements like x = y/32.
“x=y/32 means that x is 1/32 as large as y. That means that y is 32 times as large as x. So, y=32x.
The evolution of students’ understandings of reciprocal relationships of relative size is
still being researched, especially by Steffe and his colleagues [32, 60, 62, 63, 77]. The fact that
24
In the past, some people have understood examples like this to imply that we advocate solving all
problems non-symbolically. Nothing could be further from the truth. What we advocate is that rules and
shortcuts for operating symbolically should be generalizations from conceptual operations instead of
being taught in place of conceptual operations.
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this understanding happens so rarely among U.S. students makes it quite hard to research its
development. But the fact that these understandings happen so rarely is a significant problem for
U.S. mathematics education, for there is some evidence that it is expected more routinely
elsewhere [16, 43, 44]. It should be a dominant topic of discussion that U.S. instruction fails to
support its development, and the reasons for that failure should be understood in detail. At
present, we can say little more than “it doesn’t happen because few teachers and teacher
educators expect it to happen.”
The matter of students’ and teachers’ inattention to what numbers are and what it means
to operate on them is reminiscent of a discussion between Samuel Cutler [39] and John Conway
[13] in which Kutler wondered about Conway and Guy’s [13]’s treatment of fractions. Kutler
asked:
… they [Conway and Guy] give this example: 2/3 x 1/4 = 4/6 x
1/4 = 1/6. What is the logic of this? Do [they] think that the
readers have learned that the definition of the product of a/b and
c/d is ac/bd and that they have forgotten it? Do they think that the
reader knows or will investigate the justification for compounding
the ratios of numbers, or what? [39]
Conway responded:
Guy and Conway DO think, unfortunately, that readers have
learned that that is the definition of the product, which it really
isn't, except in formal axiomatic contexts. Suppose one stick is
three-quarters as long as another, whose length is two-and-a-half
inches. Then do you really think that the reason the shorter stick's
length is 15/8 inches is because the DEFINITION of a/b times
c/d is ac/bd? It ISN'T! It's because what actually happens is that
if you cut the longer stick into 4 equal quarters and throw one of
them away, the total length of what's left will actually be 15/8
inches. [13],
Kutler spoke from a presumed image that the result of multiplying fractions is determined
by definition, by an algorithm. Conway pointed out that the result of multiplying fractions is
determined by what multiplication of fractions means. In his example, 3 4 × 2 12 refers to a length
that is 3 times as long as is 1 4 of 2 21 . That is the way he turned the multiplication expression
into a thing to which it referred. It happens that the result is 15/8 not by definition, but by
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coincidence. The rule “a/b × c/d = ac/bd” is a generalization that derives from the meanings of
multiplication and of fractions in conjunction with a particular notational system for expressing
those meanings. We agree with Conway that it is misguided to portray multiplication of fractions
as being defined by a generalization, and we support the larger implication that it is misguided to
portray numbers and operations as being defined by what could be (under a more enlightened
pedagogy) generalizations from fundamental meanings.
Multiplicative numbers — a synthesis
The title of this section does not point to what we will do in it. We will not synthesize what
we’ve said about measurement, multiplication, division, and fraction. Rather, it points to what
multiplicative numbers are. Multiplicative numbers are a synthesis. We do not mean this in a
realist sense. We mean it in a cognitive sense. Multiplicative numbers become “real” when
people understand them through complementary schemes of conceptual operations that are each
grounded in a deep understanding of proportionality. At the same time, a focus on developing
these schemes will enrich students understanding of proportionality.
Most evidence points to the firm conclusion that it is rare for multiplicative numbers to
be real to U.S. students. The extent to which other countries’ educational systems and broader
culture support students’ creation of that reality is an open question, but an important one.
Nevertheless, our system would be improved were it to support students’ creation of
multiplicative numbers. We need not compare ourselves to other countries in setting goals for
ourselves.
Conclusion
We have attempted to make explicit one image of what “to understand” means in regard to
numbers and numeric operations beyond counting and whole numbers. In doing so we described
conceptual operations that might underpin those understandings. As we said in an opening
paragraph, we do not consider our work in this chapter to be definitive. Rather, we offer it as a
“sacrificial example” that, we hope, contributes to focusing scholarly discussion on what we
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intend students learn from instruction.
We have not discussed issues of symbolization, representation, symbolic skill, problem
solving, and so on. The issues we’ve raised are largely independent of symbolism and symbolic
procedure. Conceptual operations are about what one sees, not what one does, and about
reasoning that is grounded in meaning. That is not to say that “doing” is unimportant. Rather, it
addresses the common-sense issue that you should conceive of what you are trying to accomplish
before trying to do it. As we noted in Footnote 24, rules and shortcuts for operating symbolically
should be generalizations from conceptual operations instead of being taught in place of
conceptual operations. Once done, symbolic operations can become the focus of instruction. But
as a number of people have stressed, it is important that students have a rich web of meanings to
fall back on when they become confused or cannot recall a remembered procedure[6, 10, 21, 67,
70, 82].
There is a practical drawback to a scheme-based characterization of learning objectives.
It is that assessment of whether students have achieved a learning objective is more complicated
when expressing objectives in terms of schemes of conceptual operations than when expressing
them in terms of behavioral skills. When learning objectives are stated in terms of skills,
determining whether a student has achieved the objective is straightforward. When learning
objectives are stated as schemes of operations, students’ behavior must be interpreted to decide
whether it reflects reasoning that is consistent with the objectives’ development. This
complication is unavoidable. However, coming to see reasoning in students’ behavior is part of
coming to understand the scheme-based approach we’ve used, so attempts at making teachers
sensitive to students’ reasoning may have the side-effect of preparing them to think of learning
objectives in terms of conceptual operations.
In closing, we re-emphasize our motive for this chapter. We believe that conditions for
improved instruction entail an enduring discussion of what the community intends students learn.
When teachers and teacher educators lack clarity and conceptual coherence in what they intend
students learn, they introduce a systematic disconnection between instruction and learning. This
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becomes is a major obstacle to creating instruction that empowers students to think
mathematically. Note that we said that the problem is a lack of clarity and coherence in what
teachers and teacher educators intend that students learn; we did not say that there is any one
correct intention. Nevertheless, not all intentions are equally coherent or equally powerful. We
hope this chapter helps initiate prolonged discussions of what those intentions should be.
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