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If y = 3x, ™, is y greater than x? Teachers Evolving Understanding of Operations on Quantities

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Learning about mathematics entails the study of numbers. Does it really entail the study of quantities? Or should considerations about quantities belong exclusively to the realm of science education? If not, how are students to make sense of word problems, where the objects of interest are not just pure numbers? Where, if at all, is it important for teachers and students alike to interpret a symbolic expression differently, according to whether the components are taken to represent pure numbers or quantities? What sort of understanding regarding quantities would be helpful, or even essential, for K-8 mathematics teachers to learn and to include in their teaching? In this study, we analyze how teachers consider the relationships between the elements in the algebraic expression of a function as quantities, before and after online discussions in an online teacher development course. We also analyze how their answers evolved, over the course of one week, as they discussed answers with their group instructor and their peers. What are quantities? What is quantitative reasoning? There is a general consensus in the physical and medical sciences that a quantity refers to a "property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference (BIPM et al, 2012)," the reference typically being either a unit of measure or, in the case of discrete quantities, an enumerable kind of entity (atom, molecule, cell, planet, chromosome, etc.). This definition will may serve us in mathematics education contexts, provided we include quantities outside the physical sciences (e.g. price, number of children, average test score, number of solutions). In English, the term quantity has two distinct meanings (Bièvre, 2009): it may refer to the property itself (length, weight, volume, time, density) as well as the value of the quantity associated with a particular object or collection of entities (3.1 m, 7 g, 14.3 in 3 , 3.5 h 11 pencils). In some languages, these meanings are expressly distinguished (e.g. grandeur and quantité, in French). In English, one might employ quantity value for the latter case, but, by itself, the term quantity will often be ambiguous without additional qualification. Furthermore, although we will carefully distinguish between quantities and pure numbers (e.g. 44, Pi,-3.1, 7 + 4i), we cannot assume that this convention is adhered to by either teachers or students. For the present purposes, quantitative reasoning refers to reasoning about relations among quantities (as described above), whether they are generic (grandeurs) or associated with particular values (quantités). Schwartz (1996) contrasts the mathematics of quantities to the mathematics of number devoid of referents. Lobato and Siebert (2002) consider quantitative reasoning as reasoning with measurable properties of objects. And the publication, Common Core State Standards for Mathematics (CCSSM), contrasts quantitative reasoning with abstract reasoning, describing the
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If 𝒚 = 𝟑𝒙, is y greater than x?
Teachers Evolving Understanding of Operations on Quantities1
Analúcia D. Schliemann, Chunhua Liu, David W. Carraher, & Montserrat Teixidor-i-Bigas
Paper presented at the NCTM 2017 Research Conference, San Antonio, TX, April 3-5.
Learning about mathematics entails the study of numbers. Does it really entail the study of
quantities? Or should considerations about quantities belong exclusively to the realm of science
education? If not, how are students to make sense of word problems, where the objects of
interest are not just pure numbers? Where, if at all, is it important for teachers and students alike
to interpret a symbolic expression differently, according to whether the components are taken to
represent pure numbers or quantities? What sort of understanding regarding quantities would be
helpful, or even essential, for K-8 mathematics teachers to learn and to include in their teaching?
In this study, we analyze how teachers consider the relationships between the elements in the
algebraic expression of a function as quantities, before and after online discussions in an online
teacher development course. We also analyze how their answers evolved, over the course of one
week, as they discussed answers with their group instructor and their peers.
What are quantities? What is quantitative reasoning?
There is a general consensus in the physical and medical sciences that a quantity refers to a
“property of a phenomenon, body, or substance, where the property has a magnitude that can be
expressed as a number and a reference (BIPM et al, 2012),” the reference typically being either a
unit of measure or, in the case of discrete quantities, an enumerable kind of entity (atom,
molecule, cell, planet, chromosome, etc.). This definition will may serve us in mathematics
education contexts, provided we include quantities outside the physical sciences (e.g. price,
number of children, average test score, number of solutions).
In English, the term quantity has two distinct meanings (Bièvre, 2009): it may refer to the
property itself (length, weight, volume, time, density) as well as the value of the quantity
associated with a particular object or collection of entities (3.1 m, 7 g, 14.3 in3, 3.5 h 11 pencils).
In some languages, these meanings are expressly distinguished (e.g. grandeur and quantité, in
French). In English, one might employ quantity value for the latter case, but, by itself, the term
quantity will often be ambiguous without additional qualification. Furthermore, although we will
carefully distinguish between quantities and pure numbers (e.g. 44, Pi, -3.1, 7 + 4i), we cannot
assume that this convention is adhered to by either teachers or students.
For the present purposes, quantitative reasoning refers to reasoning about relations among
quantities (as described above), whether they are generic (grandeurs) or associated with
particular values (quantités).
Schwartz (1996) contrasts the mathematics of quantities to the mathematics of number devoid of
referents. Lobato and Siebert (2002) consider quantitative reasoning as reasoning with
measurable properties of objects. And the publication, Common Core State Standards for
Mathematics (CCSSM), contrasts quantitative reasoning with abstract reasoning, describing the
1 This study is part of a National Science Foundation (NSF) Math Science Partnership project (grant #0962863),
The Poincaré Institute for Mathematics Education (https://sites.tufts.edu/poincare/ ). Opinions, conclusions, and
recommendations are those of the authors and do not necessarily reflect NSF’s views.
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former as imputing a context to a problem formulated in symbolic terms and the latter as a way
of “decontexualizing” the circumstances of a particular context to a wider one by means of
symbolic formulations. In its Mathematics Practices, the CCSSM (2010) further describe
quantitative reasoning as follows:
Quantitative reasoning entails habits of creating a coherent representation of the problem
at hand; considering the units involved; attending to the meaning of quantities, not just
how to compute them; and knowing and flexibly using different properties of operations
and objects. (p.6).
Despite subtle differences in descriptions and definitions by different authors, quantitative
reasoning may be instantiated in terms of reasoning about relationships among quantities as
opposed to reasoning about pure numbers. Moreover, as highlighted by Thompson et al. (2014),
Lobato and Ellis (2002), and Ellis (2007), quantitative reasoning is fundamental for mathematics
education and to grades K-12 teacher preparation.
Quantitative reasoning in mathematics education
There are two simple views regarding the relationship between quantitative reasoning and
reasoning about pure numbers. The first view is that one learns about numbers, their operations,
and their properties and then learns to apply that knowledge to extra-mathematical contexts
involving quantities. An opposing view is that a child’s mathematical knowledge about number
is constructed upon their reasoning about quantities.
The idea that mathematical knowledge is constructed upon reasoning about quantities is not a
new one and a wealth of previous research points to the role of contexts, situations, and
quantitative reasoning in learning and development. Studies of how children develop logical
reasoning and come to understand mathematical relations point to the importance of children’s
reflections upon actions on objects as the source of logico-mathematical understanding (Piaget,
19xx). Studies of everyday reasoning show that even unschooled individuals can learn about
mathematical relations and properties as they participate in socio-cultural activities involving
physical quantities and measurement (Carraher, Carraher, & Schliemann, 1985; Nunes,
Schliemann, & Carraher, 1983; Lehrer, 2003; Lehrer et al., 1999; Saxe, 1991). Vergnaud (1994)
proposes that understanding mathematical concepts involves considering invariants, symbols,
and situations. Mathematics education researchers have emphasized the importance of worldly
phenomena and situations and agree that mathematical understanding builds upon and emerges
as a result of student thinking about relations among quantities (e.g., Freudenthal, 1983; Lehrer,
2003; Liu et al., 2017; Thompson et al. 2016) such as length, weight, time, area, volume, speed,
unit-price, density, or number of objects. Whitney (1968a, b) has proposed that quantities should
be explicitly integrated into the mathematical models used to solve applied problems. Ellis
(2007) found that, teaching grounded in quantitative reasoning produces more and stronger
generalizations than teaching focused on numerical or calculation patterns and enable the learner
to draw inferences about relationships that may not be present in the situation itself. Previous and
current standards and frameworks for mathematics education in the United States highlight the
importance of students’ work with physical quantities and the preparation of students to use
mathematics to solve every day or science problems, starting from the elementary school years.
Teaching and the mathematics of quantities
It is commonly expected that mathematics teachers are to promote quantitative reasoning among
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their students, as fundamental steps towards understanding arithmetic as well as algebraic
representations and syntactic rules. To this goal, teachers need to be aware of the mathematics of
physical quantities. However, in a study of teachers’ quantitative reasoning, Thompson et al.
(2014) has found that quantitative reasoning may present challenges to teachers. Some of
teachers’ difficulties with quantities have also arisen in the context of our work in a teacher
development program (Teixidor, Carraher, & Schliemann, 2013; Schliemann, Carraher, &
Teixidor-i-Bigas, submitted). For example, as part of an online discussion held in the early
stages of a teacher development program, teachers were asked whether one could represent the
relationship between distance and time with an arrow diagram, on a single line (Liu et al, 2017,
in preparation). We were surprised that, at the start of the discussions, only 12% of the 58
teachers in the program stated that the two variables could not be represented with arrows in a
single number line and justified their answers by saying that these were two different quantities
or that distance and time had different units. This suggests that they were not aware that units of
distance and time were fundamentally different and hence could not be added and subtracted nor
that their magnitudes could be compared.
Even though mathematics as a discipline has started as part of everyday activities and may
emerge, especially among young students, from considering physical quantities and situations,
traditional teaching of mathematics has often been dissociated from considerations about the
possible correspondence between the representations and structures of mathematics to worldly
phenomena. As a result, despite the core role of physical quantities in mathematical reasoning,
even when they discuss the multiple representations of, for instance, a linear function (table,
graph, algebraic expressions of the function), teachers and students may make multiple
connections among these representational formats without ever considering how they may
correspond to a given situation or event.
The absence of a focus on the type of quantities mathematical notation refers to prevents students
from using their own intuitions and reasoning as a basis for developing mathematical
understanding and for appropriating conventional mathematics representations and procedures.
It also leads students to attempts to solve problems by focusing on a string of pure number
computations and failure to interpret results in terms of the problem situation or question. As
middle and high school students proceed to the study of algebra and functions, the dissociation
between quantities and mathematics representations and conventions becomes even more
problematic. Then, the preparation of teachers to promote awareness of how phenomena in the
world, their geometrical and spatial representations, and the numerical and algebraic structures of
mathematics relate to each other should become a priority.
Multiplication and division are a good place to examine the distinction between number and
quantity. The relationship between operations on numbers and operations on quantities is
particularly difficult in problems in the field of multiplicative structures, as opposed to those on
additive structures (Vergnaud; 1988; Schwartz, 1988, 1996). For example, when one deals with
quantities, there are two kinds of operations, referent-preserving and referent-transforming
operations. Referent preserving operations combine two quantities with the same (or equivalent)
referents and produce a new quantity with the same referent. This is the case for addition and
subtraction. In contrast, multiplication and division, as referent transforming operations, allow to
combine two quantities, with the same or differing referents, to produce a new quantity whose
referent differs from either or both referents of the original quantities (e.g. a speed multiplied by
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time resulting in a distance, a measure in feet times a measure of inches per foot, yielding a
measure in inches; or a length times a length yielding an area).
Here we analyze how teachers consider or do not consider quantities when they answer questions
about a function expressed in algebra notation, before and after online discussions taking place
during an online teacher development course. We also analyze how their ways of answering the
question evolved, over the course of one week, as they participated in online discussions with
their group instructor and their peers.
Method
The data we analyze are part of teachers’ online work during the fifth week of the first of three
graduate level courses offered to a third cohort of teachers by the Poincaré Institute for
Mathematics Education program (see Teixidor-i-Bigas, Carraher, & Schliemann, 2013 and
Schliemann, Carraher, & Teixidor-i-Bigas, 2016). The three courses were offered online to
teachers in grades 5-9. They aimed at meaningful and deep learning of key topics in the middle
school curriculum viewed through the lens of algebra and functions and emphasized the
importance of multiple representations (verbal statements, number lines, function tables,
Cartesian graphs, and arithmetical-algebraic notation) for expressing relations among
mathematical objects and quantities in science and everyday situations.
The online activities involved a wide range of discussions, among teachers and course
instructors, about course topics (expressed in written notes, video lectures, apps, and video clips
from K-8 classroom activities), the solution of mathematical problems (presented as challenge
questions), and classroom activities planned and implemented by the teachers. Solving and
discussing the challenge questions constituted a substantial part of the courses, taking place
online over two weeks of each of the four units in a course. The Notes for the week the
discussions took place included material about the treatment of numbers versus quantities,
multiplication and division as referent transforming operations, and results of multiplication
operations when quantities were involved (see examples in Figures 1 and 2).
Figure 1: Arithmetic Operations on Quantities, from Course I notes, the Poincaré Institute for
Mathematics Education, Tufts University, 2016.
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A total of 49 teachers (14 teaching the elementary grade levels, 17 in middle school, and 18 in
high-school) and 20 coaches, special educators, or interventionists participated in the cohort we
examine here. They were from nine low-income school districts in three New England states.
Figure 2: Multiplication of quantities, from Course I notes, the Poincaré Institute for
Mathematics Education, Tufts University, 2016.
One of the six challenge problems teachers were asked to answer, justify, and discuss during the
week we analyze here consisted of the following set of seven questions:
(1) Can you add a distance to a distance? Either give an example and explain which
kind of quantity you would get as a result or justify why this is not possible.
(2) Can you add a distance to a different type of quantity (such as, a time, volume,
speed, force or weight)? Either give an example and explain which kind of quantity
you would get as a result or justify why this is not possible.
(3) Can you multiply a distance by a distance? Either give an example and explain which
kind of quantity you would get as a result or justify why this is not possible.
(4) Can you multiply a distance by a different type of quantity (such as, a time, volume,
speed, force or weight)? Either give an example and explain which kind of quantity
you would get as a result or justify why this is not possible.
(5) Can you divide a distance by a distance? Either give an example and explain which
kind of quantity you would get as a result or justify why this is not possible.
(6) Can you divide a distance by a different type of quantity (such as, a time, volume,
speed, force or weight)? Either give an example and explain which kind of quantity
you would get as a result or justify why this is not possible.
(7) If y = 3x, and x and y represent different quantities, can you say y is bigger than x?
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We posed questions 1 to 6 to invite the teachers to consider the meanings of different operations
(addition, subtraction, multiplication and division) on the same or different types of quantities.
Question 7, the focus of the current analysis, aimed at examining whether they would interpret
the question on the algebraic expression y=3x in terms of pure numbers or in terms of quantities
and how, before, during, and after they participated in online discussion with one another and
with each group’s instructors. Because multiplication of unlike quantities is a referent
transforming operation, the coefficient 3 transforms x into a different type of quantity, that is, y.
In such a case, where x and y represent different types of quantities, one cannot say y (e.g. a
distance) is greater than, smaller than, or equal to x (e.g. a time).
Results
Teachers submitted drafts of their responses to the challenge questions early on in the week and
their final answers by the end of the week, after feedback by instructors and by peers.
Differences in early and later answers would conceivably reflect how the views of the teachers
evolved over the week. It is not possible to determine the extent to which changes in answers
reflected autentic progress in understanding as opposed to mere adoption of conventions being
endorsed by peers and instructor. We examine the results with this caveat in mind.
Teachers Initial and Final Responses to Question 7
We first determined whether or not the teachers considered that x and y could represent different
types of quantities and, therefore, could not be compared. As shown in Table 1, in their initial
answers only 19% of the teachers considered physical quantities in their answers stating, for
instance, that “If y = 3x, and y and x represent different types of quantities we cannot generalize
that y is bigger than x because, depending upon the type of quantity, we could be comparing
unlike units which would have no meaning.”
Table 1: Teachers Answers at the Start and at the End of the Week
Teachers’ Interpretations of x, y
Initial Answer
Final Answer
As different kinds of quantities
13 (19%)
45 (65%)
As quantities of the same kind but different units.
4 (6%)
3 (4%)
As pure numbers
22 (32%)
12 (17%)
Not classified
4 (6%)
5 (7%)
No answer
26 (38%)
4 (6%)
Total
69 (100%)
69 (100%)
Among the remaining teachers, 6% considered unit transformations, that is unit conversions, in
their answers (“When I first did this I had inches and feet and same distance but with different
units. My conclusions were if my y was the larger unit, then yes, it will work--but if my y was
my smaller unit ---no.” Still, 32% interpreted the problem as entailing a comparison between
numbers, arguing, for example that, if x is a negative number, x is greater than y. The other 30
teachers, either did not answer the question (38%) or gave an unclear answer (6%).
By the end of the week, after reflecting upon questions and feedback by instructors and by peers,
the percentage of teachers who correctly treated x and y as different kinds of quantities in the
final versions of their written answers, noting that it made no sense to compare the magnitudes of
the different quantities, increased to 65%. Only 17% persisted in considering x and y as standing
for pure numbers. The percentage of teachers giving no answers and of those giving unclear
answers dropped to 7% in each case.
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The association between time of answer (initial versus final) and type of response (different
quantities versus pure numbers) was highly significant (Fisher exact probability < .0001).
Examples of Online Discussions and Changes on Teachers’ Responses
The following excerpts of teachers and instructors’ online postings in two of the groups
document how some teachers engaged in considering the quantities in the algebraic expression.
In examining these, we also try to identify strategies the instructors use to challenge and engage
teachers in considering variables and numbers in the algebraic expression as quantities and in
determining relationships among the different types of quantities represented as a function.
Let’s look at the discussions that took place in two of the groups where the instructors and
teachers engaged in the discussion of the answers to the question “If y = 3x, and x and y
represent different quantities, can you say y is bigger than x? Please justify your answer.”
Example 1
This group was composed by the instructor, seven classroom teachers (one in grade 6, three in
grade 7, one in grade 8, and two in grade 9), and one math school coach.
Initial answers by five teachers in this online group treated x and y as numbers. In such cases,
they noted that, if x is a negative number, x is greater than y, if x is zero, x and y are equal, and,
if x is positive, then x is less than zero (three teachers didn’t post an answer to this question early
in the week). XX of them presented the diagrams similar to those in Figure 1 to support their
answers2.
Such answers suggest that the teachers were treating pure numbers as valid examples of
quantities, and perhaps even different kinds of quantities. This assumption was contrary to the
notion of quantity we were trying to promote. The group’s instructor then posted the following
comment:
Instructor: In the last part of question 4 you look at the product purely as a product of
numbers and ignore the fact that we are talking about quantities. Would your answer change
if you think in terms of quantities?
One of the teachers answered the instructor as follows:
Teacher 1: I actually had thought of it for a very brief moment, initially thinking that we
would not be able to tell that necessarily if y is bigger given that x and y represented
different quantities. However, I dismissed that thought because we were not multiplying the
2 different quantities by each other. Then I got focused on the actual number scenario.
However, looking at it again, I am not sure. The reason I am confused is that you are
multiplying x just by a number, not by another quantity. 3 I am not sure how we would get y
of a different quantity. For example, if x is distance (miles) and it is multiplied by 3, we
would always get more miles so y would be the same type of quantity and the resulting
amount of miles (y) would always be equal to or greater than x.
2Typos in the transcriptions have been corrected.
3 It is true that the equation, y=3x +y, does not expressly associate the numeral, 3, with a quantity.
However, assuming that x and y are different kinds of quantities, the equation can only be valid if 3 itself
represents a quantity.
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Basically, when you talk about quantities of something not plain numbers it seems that the
answer would depend on the type of quantity (if it could be negative). However, when I
think about my product being a different quantity I am unsure because I cannot think about
an example of multiplying a quantity by just a number and getting a different type of
quantity. I hope that my answer/thinking is not too confusing... I'll wait to hear responses
and others thinking. I might be missing something completely here!
The instructor then focused on the possible meaning of 3 in the expression y=3x and provided
two examples for the quantities 3 could represent:
Instructor: If you multiply a type of quantity by 3 and get a different type of quantity, then
the 3 is not a plain number. For instance, if every child in your class needs to have 3
notebooks, you can think of the 3 as notebooks per child. Multiplying the 3 with the number
of children will give you the number of notebooks. Or if you walk at 3 miles per hour,
multiplying 3 by the hours will give you the miles. Try to come up with a few more
examples and think about the question again.
This led to the following response by a teacher, who elaborated on her changing perspective:
Teacher 1: Thanks, that now makes much more sense; I was stuck looking at the 3 as a
plain number! I am going to think of this some more and come up with other examples like
you suggested.
However, based on the examples that you gave me I would probably go back to when I first
considered that x and y were different types of quantities; at that point I was thinking that
we really could not say that y was bigger than x because you are talking about different
things. In one of your examples, we would be comparing notebooks and children; we
couldn't possibly order them because they are different. Even though we would want to just
look at the magnitude of the quantity they are just different things altogether! The same
would be true with your other example in which y is a quantity of miles and x is a quantity
of hours. You can't really say which is bigger because they are different. I will go back and
think of more examples now....
Another teacher then also expressed:
Teacher 2: I just read through what the Instructor and Teacher 1 were saying about [sub-
question] 7 of question 4. I did not think of it in terms of quantities at all and am having
some difficulty fully understanding the concept. The example of crayons and children does
help. I will be interested to read what others do with this problem.
Other teachers joined the discussion. By the end, all teachers in this group had expressed, with
examples, that, if x and y represented different quantities, they couldn’t be compared. Figures 3
and 4 show examples of their final answers.
“After considering quantities such as the number of notebooks per student or the number of eggs
you would need to make a batch of cookies (3 eggs per batch; where x is the number of batches
and y is the number of eggs needed), I can look at the quantities as numbers and say that I need
more notebooks than I have students and I need more eggs than the number of batches of cookies
I am making. I can’t compare the two though. I can’t say that 6 notebooks are greater than 3
students or that 9 eggs are greater than 3 batches of cookies.”
Figure 3. Final answer by a teacher.
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Figure 4. Final answer by a teacher contrasting operations on quantities to operations on
numbers.
Example 2
This group was composed by three teachers of grade 5, one of grades 6 to 8, one high school
teacher, and four special education or teachers of English learners. As we will see, these
teachers’ ideas were more varied. The instructor tried to trigger the discussion among the
teachers and let them figure things out by themselves, by asking questions to let them reflect on
their own answers and letting them compare each other’s ideas. Some teachers only wrote their
ideas in the pdf documents they added to the online forum space. The instructor cited answers in
the pdf documents, so that other teachers knew what these teachers said, even if they did not
open the documents.
Before any answer to the questions the following discussion on quantities took place in this
group:
Teacher 1: For Question #4 Quantities is being used to mean different values in parts 1-
6. My question is for part (7): are we using quantities as values or units?
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Teacher 2: I don't know. When I read the word quantities I automatically thought any
number, but now I am wondering if a quantity is always going to be a positive number.
Teacher 2 (again): In the Notes it says that a "quantity is some measurable property, like
height, weight, distance, cost, or speed." I don't know if this totally solves our questions.
Instructor: Hi Teacher 2 and Teacher 1, Yes, a physical quantity is a physical property of a
phenomenon, body, or substance, that can be quantified by measurement. A value for a
quantity is a numerical value in terms of a unit of measurement, such as 3 meters, 5 seconds.
The values for some quantities can be negative. For example, if we consider the ground
level is the position of zero, then the position above the ground level would be positive and
the position bellow the ground level would be negative.
The first posting (by teacher 2) about the main question (by teacher 2) came after the discussion
above and stated:
Teacher 2 (draft): This table (see below) proves that y is not always bigger than x. To say a
number is larger than another it has to fall to the right of the first number on a real number
line. Since -3 does not fall to the right of -1 it is not larger than -1 and therefore is smaller.
Also, 0 is not larger than 0.
x y
3
1
9
3
0
-1
-3
0
-3
-9
The instructor responded to this by clarifying to all group members that the question referred to
different types of quantities:
Instructor: Hi everyone, In part 7 of Question 4, it says "If y = 3x, and x and y represent
different quantities, can you say y is bigger than x? Please justify your answer." In fact,
what we were trying to say is "x and y represent different types of quantities".
Another teacher then responded, still just considering the values of pure numbers:
Teacher 3: Here's what I'm thinking for this part... please let me know if my thinking makes
sense. No, we cannot say that y is bigger than x because we don't know the value of it. It
could be a positive or negative integer for that matter, and there are endless possibilities.
(I'm thinking back to U1W1, knowing that there is an infinity amount of numbers between 3
and 4.) There is nothing that tells me what y equals, therefore, I have no basis to know what
the value of it is and there is an infinite amount of possibilities.
The instructor considers the last answer and takes the opportunity to raise a new question aimed
at better understanding the teacher’s reasoning:
Instructor: Hi Teacher 3, If we know that both x and y are positive, can you say that y is
bigger than x?
Following the instructor’s question, the teacher responds considering only plain numbers and the
instructor then reminds her that the question refers to quantities:
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Teacher 3: I just inputted numbers into a chart (not shown in the forum) for x and y...and
besides when x and y are 0,0, the answer is YES, y will always be bigger than x when you
use positive numbers. For example, when y=3x; 12 =3times 4 and y=3x when 6=3times 2
and y=3x; But, when you use (0,0), x and y are equal. Am I on the right track?
Instructor: Hi Teacher 3, How about if the x and y represent different types of quantities?
Can you say y is bigger than x, if they are positive?
The teacher tentatively switches her focus to quantities and gives an example, but she still
thought that one type of quantity could be bigger than another type of quantity.
Teacher 3: Rather than thinking mathematically, I thought of “x and y” as different types of
quantities and used the formula for work that we are using in science. work (y) = force (3)
times distance(x). Therefore, the work is going to be bigger than the distance, so, y will be
bigger than x. Am I on the right track?
Another teacher answers by considering unit conversion situations (although he made a mistake
in considering the conversion factor):
Teacher 4: Do you necessarily have to convert whatever one unit is to the other? For
instance, what if you have y = ounces and x = pounds. If x=5: so y=3(5), y = 15. Would you
necessarily have to then convert the 15 to the proportional ounces to pounds? If not, then
no, y is not always bigger than x. In this example, we would be saying that for every 5
pounds, you would have 15 ounces. Does this make sense?
Teacher 4 (again): Or maybe a better example would be if you were using gallons and
ounces with: x = gallons; y = ounces, millimeters or centimeters. Once again, as long as
you don't have to convert one unit to the other and are looking at it as a ratio, y would not be
bigger than x.
The instructor reminds Teacher 4 that the question refers to different types of quantities:
Instructor: Hi Teacher 4, In your example, both x and y represent the same type of quantity,
volume. How about if x and y represent different types of quantity? Can you say that y is
bigger than x?
Teacher 4: I think so! What if I substituted ounces for inches. Once again as a ratio only
you would have the x being bigger than y. I'm not sure if that is something one could do.
Apparently, Teacher 4 was still thinking about the same type of quantity with different units.
Although he said inches, he seemed to mean cubic inches, which was still the unit of volume.
The instructor drew his attention to what another teacher had said about this question to help him
understand what “different types of quantities” meant.
Instructor: Hi Teacher 4, Do you mean that, if you substituted ounces for inches (y = inches
and x = pounds), then you think that y is bigger than x? Here is what Teacher 5 said, in her
work, "If y = 3x, and y and x represent different types of quantities we cannot generalize
that y is bigger than x because, depending upon the type of quantity, we could be comparing
unlike units which would have no meaning. For example, if x = 60 miles per hour, and 3
signifies 3 hours then y = 180 miles. While the number 180 is, indeed, greater than the
number 60, we cannot quantify that 180 miles is greater than 60 miles per hour because we
would be comparing unlike quantities (distance and distance/time), which would essentially
have no meaning." What do you think?
12
Teacher 4: I would agree with her. The only way that I could find that with y=3x, for y to
be greater than x, it would have to be a completely different unit altogether. It would
therefore not have any meaning.
Teacher 4 said that he agreed with Teacher 5, but he also said: “for y to be greater than x, it
would have to be a completely different unit altogether. It was not clear what this meant. Then
Teacher 1 joined the discussion. She also considered this function as a unit conversion formula
and viewed x and y as the same type of quantity (length). In respond to Teacher 1, Teacher 5
explained her ideas again.
Teacher 1: Thanks for clarifying. That was my original question. So for example if x= 1
foot then 3x would = 1 yard. This example it does work, but when you use negative
numbers x= -1 foot then y= -3 feet, it does not work. Am I on the right track?
Teacher 5: Hi Teacher 1, I'm not sure if this is correct or not, but I think that for this
particular problem, the numbers we substitute for x and y don't matter as much as the types
of quantities that we use. For example, if we assume that x = 60 miles per hour, and 3 = 3
hours then y would = 180 miles. While the number 180 is, indeed, greater than the number
60, we cannot quantify that 180 miles is greater than 60 miles per hour because we would be
comparing unlike quantities (distance and distance/time), which would essentially have no
meaning. I hope I am not confusing the issue!
Since most teachers have switched to consider different types of quantities, the instructor tried to
let all the teachers compare different ideas from two teachers who had both considered different
types of quantities.
Instructor: Hi everyone,
Teacher 3 said: "Rather than thinking mathematically, I thought of "x and y' as different
types of quantities and used the formula for work that we are using in science. work (y) =
force (3) times distance(x). Therefore, the work is going to be bigger than the distance, so, y
will be bigger than x".
Teacher 5 said: "I'm not sure if this is correct or not, but I think that for this particular
problem, the numbers we substitute for x and y don't matter as much as the types of
quantities that we use. For example, if we assume that x = 60 miles per hour, and 3 = 3
hours then y would = 180 miles. While the number 180 is, indeed, greater than the number
60, we cannot quantify that 180 miles is greater than 60 miles per hour because we would be
comparing unlike quantities (distance and distance/time), which would essentially have no
meaning".
Who do you agree with?
Two teachers then responded in the following ways, which were different from their earlier
postings:
Teacher 2: I think that if you strictly look at the numbers then yes, y will be bigger than x in
both cases. If you look at the units they represent in each case you can't compare them
because they are unlike quantities.
Teacher 1: So, the answer to question “can you say that y is bigger than x?" would be no as
the types are different and we cannot compare two different values of different types of
quantities. Would this be accurate?
13
Some general trends
For many teachers, the word quantity may be conflated with number. After reading the question,
most teachers did not consider that x, y and 3 were quantities that described properties of objects
and had different units. Then, many answered the question by examining cases in which x and y
are either positive numbers, negative numbers, or zero. The trend of only considering pure
number was very strong, given the question was posed right after the six sub-questions on
operations on quantities.
Some teachers did consider x and y as quantities, but interpreted 3 as a pure number. That was
the case of Teacher 1 in the first group above, when she said: “I cannot think about an example
of multiplying a quantity by just a number and getting a different type of quantity.
Teacher 4 in the second group considered 3 to be a unit conversion factor, that is, x and y were
viewed as the same type of quantity but with different units. His answer that x and y were the
same made sense since he considered x and y as the same quantity expressed in different units.
Teacher 1 in the second group also considered x and y as quantities, but unlike Teacher 4, she
focused only on the size of numbers when she said: “… if x= 1 foot then 3x would = 1 yard.
This example it does work, but when you use negative numbers x= -1 foot then y= -3 feet, it does
not work.” Teacher 5’s explanation seems to have helped Teacher 1 who, at the end of the week,
realized that we could not compare the values of different types of quantities.
Teacher 5, also in the second group, gave a good answer at the beginning and her ideas and her
further explanation in the discussion did help other teachers. Teacher 4 and Teacher 1’s ideas on
considering 3 as a unit conversion factor were also valuable, because these are also important
situations for using functions.
Where and how did progress occur?
Table 2 shows that the percentage of teachers who changed from an exclusive focus on numbers
in their first postings to considering quantities and expressing that one cannot compare x and y
because they represent different kinds of quantities varied greatly across groups.
Table 2: Instructors’ Postings and Teachers Answers at the Start and at the End of the Week
Groups
Instructors’
Postings on
Quantities
Teachers’
Initial Answers
on Quantities
Teachers’ Final
Answers on
Quantities
Percent of Changes
from Numbers to
Quantities
A (N=9)
2
1
3
25% (2 out of 8)
B (N=9)
1
1
3
25% (2 out of 8)
C (N=8)
2
2
4
33% (2 out of 6)
D (N=9)
1
2
5
43% (3 out of 7)
E (N=9)
5
5
7
50% (2 out of 8)
F (N=9)
12
0
6
67% (6 out of 9)
G (N=8)
5
2
8
100% (8 out of 8)
H (N=8)
3
0
8
100% (8 out of 8)
The table also shows the number of instructor’s postings on the y=3x question. In the four groups
where fewer than half of the teachers changed from considering numbers or unit transformation
to considering quantities (groups A to D), the discussions geared to other topics and only one or
14
two questions were asked on the answers to the challenge questions analyzed here. In group H,
where all teachers progressed from a focus on numbers in the initial responses to considering
quantities in their answers, even though the instructor only posted three comments on the
answers to the questions, these were detailed and geared to eliciting teachers’ reflections from a
very interactive group, where participants often reacted to postings by the instructor and by their
peers.
Discussion
At the beginning of discussions, teachers were more likely to assess the question (if y = 3x, is y >
x?) by assuming that x and y were pure numbers (32% vs. 19%). By the end of the week, the
trend had reversed: 17% of the teachers were treating x and y as pure numbers. Fully 65% of the
teachers were evaluating the question under the assumption that x and y were quantities of
different kinds. Differences across groups may have been, at least in part, due to instructors’
ways of addressing the issue as it emerged. In the groups where instructors raised questions
aimed at eliciting reflection about relations between elements in the algebraic expression and
about how these related to quantities, most or all teachers switched to considering quantities,
instead of just pure numbers.
In part, the initial responses may have resulted from a different interpretation of the term
“quantity”, something that could be easily addressed. However, we need to keep in mind that
considerable effort was put into explaining and exemplifying what was meant by quantity, both
in the course notes and in the challenge question itself. This suggests that it’s not enough to
“clearly define the term, quantity, for the teachers”. They need to discuss their views and the
instructors’ role in these discussions are an important factor in the teachers’ development.
Whether one interprets the variables as quantities leads to very different answers to the question
“if y = 3x, (and y and x are different kinds of quantities), is y > x?” More generally, whenever
one is modeling, it is important to distinguish between numbers and quantities.
A focus on relationships between pure numbers is of utmost importance in mathematics.
However, a focus on quantification is essential in promoting students’ mathematical
understanding. A balance between the two views is certainly the ideal approach.
Students need to be given the opportunity to consider the variables and numbers in the algebraic
representation of functions as quantities much earlier and more often. This would allow for a
deeper understanding of mathematics and its representations and would prevent the view of
functions as pure numbers, a view that would be very to change after it has been settled in the
early years.
Future analysis of instructors and teachers’ online interactions will examine ways to promote
teachers’ consideration of how mathematical notation represents quantities in terms of what is
represented, and on how to engage students in considering quantities.
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