A Quantitative Reasoning Approach to Algebra Using InquiryBased Learning
Piercey, Victor I.. "A Quantitative Reasoning Approach to Algebra Using InquiryBased Learning." Numeracy
A Quantitative Reasoning Approach to Algebra Using Inquir yBased Learning
Victor I. Piercey 0
0 Ferris State University

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Cover Page Footnote
Victor Piercey received his Ph.D in mathematics from the University of Arizona in 2012. Dr. Piercey also
holds a B.A. in Humanities from Michigan State University, a law degree from Columbia University with a
certificate in international and comparative law, and a M.S. in Mathematics from Michigan State University.
He practiced law in the New York office of Weil, Gotshal, & Manges LLP for two years before returning to
Michigan for a career in Mathematics. Dr. Piercey began working as an assistant professor in the mathematics
department at Ferris State University in the fall of 2012 and is currently an associate professor. He is interested
in using his legal experience to enhance his mathematics instruction and provide students with transformative
experiences. His current project is an inquirybased sequence of courses entitled Quantitative Reasoning for
Professionals. Some sections of this course are linked with sections of freshman writing courses.
Thi s article is available in Numeracy: http://scholarcommons.usf.edu/numeracy/vol10/iss2/art4
Introduction
Quantitative literacy and quantitative reasoning courses are often considered
“less rigorous” alternatives to algebra. For example,
Hamman and Kung (2017)
express a concern that quantitative literacy courses are used in place of algebra,
and this practice may unintentionally impede equitable access to highquality
mathematics. Their commentary is based in part on Moses and Cobb (2002), in
which access to quality instruction in algebra is cast as a civil right. While
Hamman and Kung (2017)
are careful to point out that quantitative literacy and
algebra need not and should not be disjoint, their empirical observations illustrate
a common, unstated assumption in the community – that quantitative
literacy/quantitative reasoning is the “nonalgebra” curricular option. This
assumption needs to be challenged. For example, many quantitative reasoning
textbooks include topics involving linear and exponential modeling
(e.g., Bennett
and Briggs 2014)
. Steen (2001) stated that quantitative literacy and reasoning
required knowing “more than formulas and equations,” an indication that we are
actually talking about a larger, and deeper, domain of thinking.
Gaze (2014)
proposes that quantitative reasoning, with an emphasis on proportional reasoning
and modeling, should enhance algebraic reasoning.
On the other hand, numeracy, quantitative literacy, and quantitative reasoning
are often characterized by a focus on numerical, not algebraic, information. Steen
(1999) himself emphasized understanding numerical information that surrounds
us in what he called a “datadrenched society.” Zillman et al. (2009) place the role
of numerical data presented in the news at the center of quantitative literacy. The
Association of American Colleges and Universities (2010) in its VALUE rubric
describes quantitative literacy as “an athomeness” with numbers. Others, such as
Geiger et al. (2015)
consider quantitative literacy to be essential to citizenship and
participation in a democracy.
In the following, we ask what the proper relationship is between quantitative
reasoning and algebra. More specifically, we ask how algebra and quantitative
reasoning may overlap, what a curricular module implementing this overlap could
look like, and what impact such a curricular module may have on students’ basic
algebraic skills. Throughout, we should observe that facility with numerical
information and algebra need not be mutually exclusive. In fact, these skills can
support one another, as
Gaze (2014)
envisions.
As an illustration of what a quantitative reasoning approach to algebra could
look like, compare the following two problems:
Problem 1: Solve
= 1 − for A.
Problem 2: The following formula tells you the expected rate of return on a new stock
issue:
=
1
0(1 −
)
+
The variables are defined in the attached supplement.1 Note that the variable F is how
much it costs to float the new issue. Typically, consultants, such as bankers and lawyers,
help you to put a new stock issue on the market. They are paid a percentage of the total
proceeds. This percentage is what F stands for.
Suppose you are auditing an investment bank and they have all of the data for their 10
most recent stock issues, except they are missing the floating costs! Since this is their
fee, you want to make sure everything is as it seems.
The data are in a spreadsheet. Program Excel to calculate these floating costs for you.
The first problem is from a standard beginning and intermediate algebra textbook,
and the second is from my course, which will be described in more detail below.
Both problems require solving an equation for a specified variable. In the first
problem, we (the students and I) don’t know what the variables stand for, nor do
we know why we would want to manipulate the formula. The second problem is
motivated by a contextual need related to a realistic scenario – auditing. The
formula needs to be solved for F in order to find the missing data, and Excel is an
integral part of the problem. What’s more – the algebra is considerably more
difficult. As a bonus, students learn something about initial public offerings and
transaction costs. Note that in the course materials described in this paper, there
are many exercises that are not as well motivated – but the vast majority do use
formulas with actual meaning and defined variables, and there are enough
contextual problems such as the floating cost calculation of Problem 2 that
students working the exercises should be able to imagine when they might want to
solve the exercises in context.
The relationship between quantitative reasoning and algebra is more than an
abstract concern. In generaleducation reform, current pathways initiatives for
colleges and universities (such as Quantway and Statway published by the
Carnegie Foundation2 and the New Mathways Project sponsored by the Charles
A. Dana Center3) envision three pathways through mathematics: a pathway to
1 The supplement is omitted from this paper. The variable is the expected return on an initial
public offering (IPO) of stock where 1 is the expected dividends to be paid in the first period, 0
is the initial price offered for the stock, F stands for the flotation costs (described in the problem),
and g is the expected growth rate of the stock, which is assumed to be constant.
2 https://www.carnegiefoundation.org/inaction/carnegiemathpathways/ (accessed by author on
Jan. 20, 2017)
3 http://www.utdanacenter.org/highereducation/newmathwaysproject/ (accessed by author on
Jan. 20, 2017)
calculus; a quantitative reasoning pathway involving algebraic modeling but not
necessarily manipulations; and a statistics pathway. However, occupational
students, such as business students, do not fit neatly into one of these three
pathways. In particular, they need both quantitative reasoning skills and the
ability to manipulate algebraic formulas4
(e.g., see Rosenbaum 1986, Ely and
Hittle 1990, Anderson, et. al. 1994, Ballard and Johnson 2004, Green et al. 2007,
and Raehsler et al. 2012)
. Business programs are not the only programs that need
a mix of quantitative reasoning and algebra. When we drill down to the
mathematical needs of students in STEM programs, we find a basic need for
quantitative literacy. When the Mathematics Association of America (MAA)
committee on Curriculum Reform across the First Two Years (CRAFTY) held
discussions with partner disciplines about the mathematics needed for their
students, much of what was addressed could be considered quantitative literacy or
reasoning
(Ganter and Barker 2004, Ganter and Haver 2011)
. These discussions
included faculty in disciplines such as physics, whose students are often assumed
to possess a high degree of mathematical competence.
This discussion raises two questions: how quantitative reasoning fits into
traditional algebra and calculus, and how traditional algebra fits into quantitative
reasoning. In addition to
Gaze (2014)
,
Karaali (2008)
shows how quantitative
reasoning can be embedded into calculus, while
Catalano (2010)
and Tunstall and
Bossé (2015) illustrate how to include quantitative reasoning in college algebra.
We will focus on the second question, fitting algebra into quantitative reasoning.
Relative to this question, Wallace (2011) describes stages of cognitive
development in the learning of algebra using a framework of
Harper (1987)
, and
she points out that appropriate attention to these stages can enhance quantitative
literacy. Wallace’s perspective applies best to K12, as early contacts with content
should be designed according to cognitive development.
In this paper, we will look at what algebra looks like in a quantitative
reasoning course for college students, most of whom struggled significantly with
algebra in high school. I will present my analysis in two parts. The first part will
be the theory and background that informed the design of my curricular materials.
I will share my own perspective on what we mean by “algebra” and “doing
algebra,” which will then be compared with how Karaali et al. (2016) define
numeracy, quantitative literacy, and quantitative reasoning. These “theoretical”
perspectives inform the design of the course materials and are necessary to
4 As an example from another time under different circumstances, Service (2000) points out that
after the Russian Revolution and Civil War, Lenin was anxious to help the workers and peasants
of the emerging Soviet Union become literate, but was also concerned with numeracy. The need
to be numerate was driven by the desire for workers to run state enterprises and for peasants to run
agricultural cooperatives.
understand how the lessons are constructed and scaffolded.5 In the second part,
using part 1 as background, I describe a sixweek module that was implemented
in a course for business students. I will end with data comparing student
performance in the quantitative reasoning course to those in traditional algebra
courses. The evidence of student success illustrates the power that quantitative
reasoning can bring to algebra. The students who took the course had been placed
into developmental algebra 1, but the percentage of students successfully
completing standard algebra problems is comparable to the performance of
students in precalculus.
The data have limitations. Despite these limitations, I hope that readers of
this paper will start conversations with their colleagues about the relationship
between quantitative reasoning and algebra and how the two fit together and
support one another.
Part 1: Theory and Background
To begin, we should ask ourselves what algebra is. The question not only helps
clarify what we are talking about, but my perspective on it – namely, the nature
and essence of algebra – informs the design of the curricular materials and how
they are described in Part 2.
A Personal Perspective on Algebra
Most high school graduates have a sense of what algebra means. After all,
algebra is the heart of the high school mathematics curriculum. However, what if
someone were asked to define algebra? What does it mean to “do” algebra? Some
might refer to topics in a table of contents from a textbook, while others may
reply that they “know it when they see it.” In order to consider an approach to
algebra based on quantitative reasoning, we should understand what we mean by
algebra and algebraic manipulations. What follows is my perspective.
Algebra as Communication
My first contention is that algebra is a form of communication. Algebra expresses
mathematical “recipes” in a way that extracts the essence of the mathematics. A
recipe tells you the ingredients, what to do with the ingredients, and the order in
5 The design was also inspired by a naïve understanding of APOS theory, a theory of cognitive
development. The name is an acronym for the four stages of cognitive understanding of a
mathematical idea in increasing depth: action, process, object, and schema. This is a very deep
theory that involves its own research methodology, and I am not familiar with the details. The
interested reader should consult Dubinksy 1991,
Dautermann 1992
,
Asiala et al. 1996
, and Arnon
et al. 2014.
which to complete each step. Algebraic equations tell you the same thing. The
variables are the ingredients, the operations are the directions, and the proper
sequencing is determined by the order of operations. For example, consider the
“future value” compound interest formula as written in business texts:
=
The “ingredients” are the variables for the amount of the initial investment (the
“present value” – PV), the interest rate (r), and the amount of time the investment
is held (t).6 Following the order of operations, the “directions” are to add 1 to the
interest rate, raise the sum to the tth power, and multiply the result by the present
value. The end result that we have “cooked” is the future value (FV) of the
investment at time t. To be certain, I am assuming a particular type of algebraic
equation (namely, a formula), but the logic can be adapted to other algebraic
forms such as implicit equations, parametric equations, or inequalities.
This perspective is wellsuited to quantitative reasoning. As noted in Karaali
et al. (2016), communication is a common element in scholars’ definitions of
numeracy and quantitative literacy. Treating algebra as communication raises
interesting questions for class discussion, such as appropriate purposes and
audiences for algebra. For example, algebra is critical if a spreadsheet is your
audience. In addition, implementing this perspective in class helps students to
build meaning out of what typically appears to them as arbitrary symbols. In the
materials described in Part 2, one activity involves students finding a way to
program Excel to calculate how efficiently a fast food franchise is handling its
fries based on data available to shift managers at the end of the night. They have
to figure out how to measure “fry efficiency” and how to communicate this
quantity to Excel (which requires algebra). In a later exercise, students reverse
this process and deconstruct a complicated formula for calculating the declining
balance depreciation expense7 into a list of instructions and create a form
capturing the process. Through this approach, students come to understand that
algebra, as I like to say, is “codified arithmetic” – connecting numerical facility
with algebraic facility.
The Algebra Diamond
The advantage of an algebraic representation of a mathematical relationship
between quantities is precision. This precision allows us to perform
manipulations that help us look at relationships in a different way, for example by
solving for one of the variables. This argument suggests that for students to be
able to make full use of the power of algebra means that they should master the
6 Assume t is in units of time that correspond to a single compounding period and r is the interest
rate for a single period.
7 The periodic expense recorded when exponential depreciation in value is assumed.
art of manipulating algebraic representations (“doing” algebra). While the
discussion below is focused on algebraic formulas (based on the needs of the
students for whom my course was designed), the ideas can be adapted to other
forms of equations and inequalities.
Solving Moves. There are two broad categories of algebraic manipulations:
“solving moves” and “simplification moves.” Solving moves are moves that are
used to solve equations or solve formulas for a specified variable. Solving moves
are the moves one would make when solving problems of the form:
Solve for r:
=
The reason for solving a formula for a specified variable is to switch the roles of
independent and dependent variables (or even make a parameter a dependent
variable). In this example, we are asked to solve the future value formula for the
interest rate. This revised formula would provide a criterion to evaluate savings
opportunities, such as savings accounts and certificates of deposits, in a situation
where we know how much we have available to deposit (PV) and want the
account balance to reach a predetermined target (FV) within a certain time frame
(t).
The underlying approach to solving moves is what I call “reverse and invert”:
reversing the order of operations in the encoded instructions and inverting. In the
example, since the last step of the formula is “multiply by PV,” the first step when
solving this formula for r is “divide by PV.” Next we invert the exponentiation by
taking a tth root, and finally we subtract 1:
=
− 1
We should note that given the nature of this problem, we are interested only in
positive, real roots. The fact that an inverse operation may result in multiple
solutions doesn’t concern us.
In addition, we should also note that the order in which we apply inverse
operations sometimes has to be adjusted on the fly. For example, what if we
started with this formula for r and wanted to solve for PV? Without getting into
details, the basic idea is that we would have to “chunk” steps together and solve
for intermediate variables. There are steps that I call “prefabricated” moves –
steps that collapse individual moves together – that help with the procedure.8 A
8 An example of a prefabricated move is applying the quadratic formula, which wraps up a
sequence of moves in the process of completing the square followed by a final sequence of inverse
operations in reverse order. Completing the square includes many “simplification moves.”
relatively simple example of a prefabricated move is what I call “the swap.”
Consider an equation such as:
=
We can solve this equation for C by “swapping” A and C. This “swap” is the
combination of multiplying both sides by C, then dividing by A.9
Simplification Moves. The other main category of manipulations consists of
“simplification moves” – moves that are used to rewrite one expression with an
equivalent expression. Factoring falls into this category, as well as performing
operations with rational expressions, or using the laws of exponents.
Simplification moves often help us see through clutter in algebraic expressions or
to rearrange the way an expression is written in order to make solving moves
more useful. At its most mechanical, simplification moves involve substitution
into identities. Simplification moves are rarely (if ever) used for their own sake
(outside of the classroom). Typically there is some strategic advantage for a
particular move: e.g., making it easier to program a formula into Excel; making it
easier to solve for a specified variable. The overall task determines whether the
result is actually “simpler” or not.
The distinction between simplification and solving moves is important
because the “rules” for solving and simplification moves are different. Solving
moves allow us to change the values of the expressions involved, but we must do
so consistently on both sides of an equation or inequality. It follows that in order
to use solving moves, we have to be working with an equation or an inequality in
the first place! Simplification moves, on the other hand, apply only to an
algebraic expression. Hence we can apply simplification moves to an expression
in isolation or on one side of an equation or inequality. The “rule” here is that we
cannot change the value: what results must take on the same values as the original
expression. In my course, I illustrate this dichotomy by comparing how two
different transactions affect a balance sheet – a financial statement in which all of
the assets are on one side, the liabilities and equity are on the other side, and the
two totals must be equal. One transaction involves purchasing an asset for cash –
exchanging one asset for another – which is like a simplification move. The other
9 There is a deeper reason that the swap works. The equation = is equivalent to = since
division inverts multiplication. With this multiplicative version of the equation, we can divide by
either A or C. This is because multiplication is commutative – we can multiply A and C in either
order. That is not the case with subtraction and division. Note that students at this level think of
addition and subtraction (as well as multiplication and division) as different, but related,
operations. If we think about the expression as B times 1/C instead of B divided by C, we would
take a different approach. Every year when I revise the course described below, I debate with
myself how hard I should push one or the other approach.
involves borrowing money – adding the same amount to both the assets and the
liabilities – which is like a solving move.
Synthesis. Problems that are more sophisticated involve a combination of both,
such as the following:10
Solve for K:
=
+ ( −
)
For this problem, at some point we will need to factor out K.
Putting all of these ideas together, the foundation for algebraic manipulations
is an understanding of algebra as communication along with the ability to
construct or to read (deconstruct) algebraic expressions. On this foundation, we
build solving moves and simplification moves, and then bring them together.
This scheme is represented in Figure 1, which I call the “algebra diamond.”
Simplification
Moves
Solving Moves
Meaning of
Formulas
Synthesis
Broadly speaking, my perspective is that algebra is a form of communication
wellsuited to the manipulations organized in the algebra diamond. To be sure,
10 Makeham’s formula for the price P of a bond. The variable K is the present value of the
redemption amount of the bond, r is the coupon rate of the bond, j is the marketbased interest rate
used to value the bond, and F is the face value.
there are many topics in algebra textbooks that do not fit this description – such as
graphing functions. Strictly speaking, those are topics in other mathematical
domains, such as analysis. I do not mean to suggest that we should avoid
teaching those topics in a class called “algebra,” but I think it is helpful to
acknowledge, at least to ourselves, that the term “algebra” may be more limited
than textbook and course content may suggest..
Difficulty Levels
My perspective on algebra has led me to hypothesize four difficulty levels for
solving formulas for a specified variable. The hypothesized levels are as follows:
1. Beginner: Can solve multistep equations for variables by applying inverse
operations in reverse order (“reverse and invert”).
A beginner could solve the following:
2. Intermediate: Can solve problems requiring a combination of reverse and
invert with “prefabricated moves.”
An intermediate student could solve the following:
Solve for Q:
=
Note that to apply the “reverse and invert” strategy to a problem such as
this one requires us to think of taking reciprocals as an operation. This
concept is not unreasonable in and of itself, but it may be confusing for
students.
3. Advanced: Can solve problems requiring a combination of reverse and
invert with prefabricated moves in which we may have to treat expressions
as objects to be manipulated. An advanced student can solve the
following:
Solve for W:
This problem can be solved with two “swaps.” First, swap E for
to
obtain
= 12 −
one variable for another, but rather one variable for an expression (12–E).
Treating that expression as an object to which we can apply inverse
operations is what differentiates the third level from the second level.
. Then swap W for 12–E. The second swap is not just
4. Mastery: Can solve problems requiring a combination of reverse and
invert, prefabricated moves, and simplification moves in which sometimes
we have to treat expressions as objects to be manipulated. A master could
solve the following problem:
Solve for K:
= − ( + )
So how does algebraic manipulation fit with quantitative reasoning? We turn
to this question next.
Quantitative Reasoning and Algebra: Does the
Shoe Fit?
To me, one hallmark of quantitative reasoning is the solving of contextualized
problems using mathematical concepts. As such, an approach to algebraic
manipulations, if couched within authentic and realistic contexts, can certainly fit
well with quantitative reasoning. Moreover, an approach to algebra as
communication that treats algebraic manipulations within the framework of the
algebra diamond encourages purposefully knowing what you are doing and why
you are doing it, another aspect of quantitative reasoning.
However, we can make the connection between quantitative reasoning and
algebraic manipulations more precise. Karaali et al. (2016) identified four
dimensions to distinguish numeracy, quantitative literacy, and quantitative
reasoning, In the process, Karaali et al. (2016) established a hierarchy of
increasing sophistication: numeracy at the lowest level, quantitative literacy at the
next, and quantitative reasoning at the highest level. The four dimensions are the
quality of the desired outcome (habits of mind vs ability vs skills), knowledge
domain (arithmetic, mathematics, statistics, for example), a display of expertise
(such as appreciate, communicate, reason), and the use of context. We can
arrange their defining characteristics in Table 1.
Examining Table 1 reveals some patterns. The “quality” with which students
interact with mathematical information grows from basic skills to habits of mind.
The knowledge domain grows to include data, and then statistics, which I take to
include inferential statistics. In the row for the “display of expertise” dimension,
we see action verbs describing what students can do. These verbs accumulate,
and as we move from numeracy to quantitative literacy to quantitative reasoning,
the additional verbs come from higher levels of Bloom’s taxonomy. The
contextual dimension does not change between quantitative literacy and
quantitative reasoning, with numeracy, notably, believed to be insufficient for
active citizenship.
Let us translate the first three dimensions to algebra. We do the translation in
Table 2, by showing how each dimension applies to algebra: first, in the level of
numeracy; next, in QL; and finally, for QR. We can think of the contextual
dimension as involving algebraic problems within practical contexts, as described
in some detail in Part 2.
Applications of the Dimensions of Numeracy, QL, and QR to Algebra
Applying the framework from Karaali et al. (2016) yields a robust and
scaffolded approach to algebra that also aligns with the sense that the words
“numeracy,” “quantitative literacy,” and “quantitative reasoning” themselves
suggest. Note that the algebraic manipulations I describe in the context of the
algebra diamond align best with “numeracy.”
Part 2: Implementation
The value of the theory in Part 1 manifests in how it is implemented in the
classroom. In this second part of the paper, we describe the course materials, both
in general and in detail, that were inspired by the theory and background. These
course materials illustrate what a quantitative approach to algebra looks like in
practice. The materials are followed by data that show how the skills of students
who have been exposed to this approach compare to those who have taken
courses in the traditional algebra and calculus sequence.
Course Materials
The course materials comprise a sixweek module, which is split into two units.
The algebra diamond and other concepts in Part 1 form the “backend” –
embedded in the design but not necessarily shared with students. The materials
were written for college business students who place into a developmental
algebra1 course. Almost every lesson involves at least one problem or exercise
grounded in contextual meaning. However, the reader will note that as the
module progresses, many of the exercises are not couched in a scenario. Rather,
most exercises involve manipulating business formulas that themselves have
meaning (with variables defined in the supplement) that is not itself addressed
within the specific problem. There are a few formulas that were cooked up but
have no apparent meaning. In the discussion below, variables are defined (mostly
in footnotes) for every formula that has meaning,
Overview of the Course
The course that these materials come from is called Quantitative Reasoning for
Professionals (QRP). It is a twosemester sequence where each semester carries 4
credits. Students who place into developmental algebra1 start with the
firstsemester course, and those who place into intermediate algebra take only the
second course. Each section has between 20 and 25 students. I designed QRP and
wrote the materials. The sequence is delivered at Ferris State University, a
mediumsized, public institution in a rural area of central Michigan. So far,
besides myself, two tenuretrack faculty and one adjunct faculty have taught the
sequence. I revise and distribute the course materials as a course pack every year.
The description below represents the third edition; the fourth is being prepared.
The best description of the content of QRP is as a hybrid quantitative
reasoning and algebra course. Business students will be taking accounting and
finance courses that require stronger algebraic reasoning than a typical
quantitative reasoning course would offer. On the other hand, at Ferris State,
business students do not take a business calculus course and do not need much of
what is included in typical algebra courses. Rather, they need to be able to reason
with data and quantitative objects. Consequently, a quantitative reasoning
approach that includes algebra is called for, and the courses were designed as an
answer to that call. The course content is organized into units as indicated in
Table 3.
4. Linear and Nonlinear Analysis
The materials described in this paper are used in the third and fourth unit of
the first semester. As implied by Part 1, if we are being careful about our terms,
the students who are successful in Units 3 and 4 would be described best as
“numerate” in algebra. In the second semester, the level at which students are
expected to use algebra mostly fits the “quantitative literacy” characteristics
described in Part 1, but by the end we also reach the “quantitative reasoning”
level. For example, at the end of the course, students use scatterplots with various
combinations of linear and logarithmic scales to select an appropriate model for
selected data sets, run appropriate regressions to estimate those models, use those
models to make predictions, and assess the strength of their assumptions. We
share some sample problems from the second course below.
The Role of Excel
The course materials use Excel heavily, both to motivate the use of algebra and to
help students understand the computational directions in a given algebraic
formula. Spreadsheets are known to enhance quantitative reasoning and have been
implemented in a variety of courses as noted in numerous papers in this journal
(e.g., Steele and KiliçBahi 2008, Gaze 2010, 2014, Vacher and Lardner 2010,
Frith 2012, Lehto and Vacher 2012, Mayfield and Dunham 2015, Ricchezza and
Vacher 2017)
.
Algebra instruction has been enhanced with Excel by others, with varying
degrees of success.
Friedlander (1998)
used Excel in a beginning algebra course
at an Israeli middle school and found that the ability to “click” on a cell reference
for a variable formed a cognitive link between arithmetic and formal algebra.
Stephens (2003) found that students were unlikely to take advantage of
extracredit assignments using Excel, but those who did so enjoyed the assignments.
Neurath and Stephens (2006) found that although using Excel in a high school
algebra course led to only a slight improvement in student achievement, it helped
narrow the gap between the high and low achievers and led to improved student
attitudes. Lehto and Vacher (2012) found that the need to learn to use Excel was
not a barrier to learning course content, but also that implementation success
depends on the instructor’s pedagogical perspective.
In this course, we begin using Excel at the beginning of the second unit.11 By
the time the students encounter the algebra units, they have developed comfort
and facility with the basic functionality of Excel. At the beginning of Unit 3, we
introduce algebraic formulas as tools to “process” data – transforming raw,
observable data into a measurement that is useful for evaluation or decision
making. The raw data are found in an Excel spreadsheet. Students have to
construct the formula they need, and then “program” Excel to use that formula.
The ability to “copy and paste” or simply “drag” the formula to the rest of the
column reinforces the idea that algebra simply codifies an arithmetic computation.
As the course progresses, students have a formula but have to solve it for a
different variable to find the missing data in their Excel spreadsheets. The “copy
and paste” or “drag” functionality reinforces the idea that when we solve a
formula for another variable, we are solving infinitely many onevariable
equations simultaneously.
As mentioned in Part 1, algebra is a natural mode of communication when
working with Excel. Anecdotally, I have noticed that students like to start with the
data in the spreadsheet before working with pencilandpaper. This observation
suggests to me that they find the spreadsheets motivating. I have noticed that as
students begin to work through this module, several students report out their
formulas using cell references as their variables. This tendency appears to
illustrate the role
Friedlander (1998)
assigned to Excel in students’ developing
conception of algebra. With guidance, this approach evolves to using standard
mathematical variables. This evolution promotes a discussion of what a choice of
variable communicates to your audience, whether Excel or a human audience.
InquiryBased Learning
The course materials are designed for inquirybased learning (IBL). Yoshinobu
and Jones (2012) define IBL as an “instructional paradigm” in which
Students actively participate in contributing their mathematical ideas to solve problems,
rather than applying teacherdemonstrated techniques to similar exercises. The instructor
facilitates student progress, ensuring that the class can move toward increasingly
sophisticated ways of thinking.
11 In the fourth edition (of my course materials), in its early stages as of this writing, we are
starting much earlier with Excel, introducing it at the fifth class meeting.
The materials in the course consist of “explorations” – guided, inquirybased
lessons. Part of the goal of the course is for students to solve novel problems, and
IBL is a tool that has shown some success in this regard
(Housman and Porter
2003, Kwon, Rasmussen and Allen 2005, Smith 2005, Chappell 2006, Rasmussen
and Kwon 2007, Selden and Selden 2013, Kohan and Laursen 2014, Ellis, et al.
2015)
. In particular, welldesigned IBL courses can help students learn to solve
problems independently without being given a “template” or a procedure for
every problem type, thereby improving problemsolving skills and reversing
unhealthy beliefs about mathematics (Schoenfeld 1988, Yohsinobu and Jones
2012). When students learn in this way, they begin to tap into their reasoning
skills and rely less on incomplete or incorrectly remembered procedures (Stigler
et al. 2010).
The classic version of IBL is the Moore Method, named for R.L. Moore12
who was a mathematician at the University of TexasAustin from 1920 to 1969.
The Moore Method, applied to upper division or graduate mathematics courses,
involves providing students with definitions only, and the students fill in the
examples and all of the proofs themselves, giving presentations to the class and
subjecting their work to the criticism of their classmates. Moore himself did not
allow students to speak with one another nor consult any materials as they did
their work, but most practitioners today relax these rules, applying the “Modified
Moore Method”
(e.g., Jones 1977, Chalice 1995, and Mahavier 1999)
.
Applying the Moore method to a freshman course for students who place into
developmental algebra may be a bit daunting. Rather, our approach to IBL for
this population involves more “guided inquiry” – that the explorations involve
carefully sequenced questions designed to start from prior student knowledge and
lead students to some new concept, and then ask students to apply that concept to
a new problem.13 In addition, the explorations are contextualized. Most are
rooted in authentic, business applications. Applications are used not only to
motivate students, but also to situate learning in an environment where we expect
meaning to be constructed and concepts to be transferrable to other courses. To be
sure, in the spirit of inquiry, some explorations are openended. For example, the
first exploration in Unit 3 (described below) asks students to come up with a
mathematical process to measure “fry efficiency.” The stakes in these
12 I acknowledge that Moore’s intellectual heritage comes loaded with his history of exclusionary
practices, including refusals to allow African American students to take his classes. Recognizing
Moore’s contributions to teaching and learning mathematics should in no way be considered an
endorsement of his exclusionary practices.
13 This design approach resembles, but does not precisely emulate, the cycle of learning that is part
of the theory underlying ProcessOriented Guided Inquiry Learning (POGIL) (e.g.,
Bénéteau et.
al. 2017
). Current revisions for a fourth edition of the course materials are bringing the design
closer in line with POGIL.
explorations are low, and students are encouraged to try different approaches,
reflect on those approaches, take risks, and pose questions.
Typically, we start class by briefly introducing any mathematical definitions
and the business context, after which students work through the explorations in
small groups. The instructor circulates around the room, checking on student
progress and responding to questions. Every five to ten minutes, the class pauses
to discuss what they have done and present their work, before returning to
smallgroup work. Sometimes the presentations are formal, with students walking
through their work at the board. Other times we just share our ideas from our
seats. While we have not measured the impact on “soft skills” such as
communication and collaboration, we have anecdotally observed growth. For
more detail concerning the use of inquirybased learning in this course, see
Piercey and Militzer (in press).
A deeper understanding of algebra and development of the framework
described above is compatible with IBL. Especially because we are teaching
students who have been introduced to algebra before, developing a framework
using constructivist approaches helps activate prior knowledge, correct
misconceptions, and put that prior knowledge into a useful organization that
students can access to solve problems.
Unit 3 Explorations
The explorations involving algebraic manipulations are in Units 3 and 4 in the
first course. The third unit (Table 4) lays the necessary infrastructure for the
algebra diamond from Part 1 (which, as mentioned above, is embedded implicitly
in the design). This unit is designed to guide students to look at algebraic
representations as lists of instructions with variables. As mentioned above, the
idea is to motivate algebraic formulas as tools to “process” data. The fourth unit
(Table 5) leads students around the rest of the algebra diamond.
Explorations 3.1 and 3.2 deal with “fry efficiency” at a fast food franchise.
The students are given an Excel spreadsheet with data that a manager would
collect on a nightly basis. The data include measurements such as the number of
fry products sold (read off a receipt) and inventory counts. The students’ job is to
figure out how to use these data to measure how efficiently that franchise is using
their fries, and this “process” (it is just a formula) has to be programmed into
Excel. In effect, students are “algebrafying” the way they use data, and Excel
forces them into using symbolic representations. The difference between 3.1 and
3.2 is how much information is given. For 3.1, students are given nothing but the
Excel sheet. This problem is open ended, and we do not necessarily expect
students to complete it; rather, we want students to have some experience tackling
something unscripted and reflect on the process. Some students completely solve
the problem by the end of 3.1, while others are able to make progress on one
small piece of it. In 3.2, we give them a “form” that is used by managers to
compute fry efficiency each evening, and students translate that form into an
algebraic formula. Students respond well to this exploration due to its realism.
Indeed, I got the idea from my own experience as a fast food manager.
The fry efficiency explorations are followed by an exploration in which
students reduce the 1040 EZ tax form into a formula that they can program into
Excel. We limit ourselves to the lowest tax bracket in force (at present, the 10%
bracket). The purpose of this exploration is to take what students learned with fry
efficiency and apply it in a different context, allowing it to sink deeper. This
example is dealt with further in the next two explorations (skipping for the
purpose of discussion Exploration 3.4, which is the last in a series of studyhabits
explorations embedded in the course14). Exploration 3.5 reviews order of
operations and applies it to the operations involved in calculating taxes. At first,
students find it counterintuitive that the steps that we need to apply in order to
calculate our taxes do not follow order of operations. This cognitive dissonance
raises the need for grouping symbols and explains their importance. In
Exploration 3.6, the distributive property is used to investigate the difference
14 Including study skills explorations was inspired by the design of the Dana Center New
Mathways Project (NMP). In NMP, the foundational mathematics course is linked with a study
skills course. See http://www.utdanacenter.org/highereducation/newmathwaysproject/
(accessed by author on Jan. 20, 2017). We did not have the luxury of doing so with QRP, so we
embedded a handful of study skills explorations into the course which culminate in a project
requiring students to use data to assess progress toward a selfidentified goal. Based on our
anecdotal observations, we are not convinced that our approach to implementing this idea has been
successful to date.
between a deduction (which is applied before the tax rate) and a credit (which is
applied after the tax rate).
In Exploration 3.7, we introduce the difference between conditional equations
and identities. The main purpose of this exploration and those that follow are to
help reduce student errors such as “distributing exponents” (for example,
asserting that ( + )2 = 2 + 2). In order to go beyond the few examples that
there is time to address, we focus on helping students to look for counterexamples
when they sense doubt about a particular move. This habit of mind is based on an
important point about mathematical argumentation: examples can be used to
prove that an equation is not an identity, but not to prove that an equation is an
identity.
Explorations 3.8 – 3.9 use counterexamples and the distinction between
conditional equations and identities to probe the order of operations further. In
particular, we look closely at alternative grouping symbols, such as radicals,
fraction bars, and absolute value bars. Following numerical calculations with
questions about algebra help students learn how these symbols fit into the order
of operations. These explorations are frequently turning points for students – the
relationship between algebra and arithmetic becomes clearer.
Explorations 3.10 – 3.11 are about reading formulas and bring this unit full
circle. First, we address identifying the dependent variable from context.
Students are given the future value formula:
=
Students describe scenarios in which each of the given variables is the dependent
variable. This exercise becomes useful later when we want to manipulate
formulas. In 3.11, students are given formulas and asked to reduce them to a list
of instructions. This activity is the reverse of what we started the unit with –
turning instructions into formulas. At the end of 3.11, students are asked to make
a form for computing the declining balance depreciation expense:15
=
∙
1 −
−1
This exploration turns the fry efficiency exploration on its head.
Exploration 3.12 was created because students had trouble connecting the
notation for radicals to their role as inverse operations for exponents (in the
limited case when we want to find an unknown base). Students are encouraged to
create their own notation first, and then compare to the established notation.
15 In this formula, DBDE is the periodic expense for depreciation when using the declining balance
depreciation technique (in which the value of the asset depreciates exponentially), CST is the
initial cost of the asset, EUL is the estimated useful life in years, PM is a preselected percentage
multiplier, and n is the age of the asset in years.
Exploration 3.13 closes this unit by having students explicitly examine the
process of generalization and abstraction that takes place when we pass from
identifying a common set of calculations to an algebraic formula. This
exploration serves to wrap up and consolidate the learning gains from Unit 3.
Unit 4 Explorations
We work specifically with manipulations in Unit 4 (Table 5). Students who were
successful in Unit 3 are at “home plate” in the algebra diamond from
Part 1. We first proceed to the right – to solving moves. We then go back and
proceed to the left – to simplification moves. We conclude by bringing the two
together. Throughout the unit, almost every formula that the students are asked to
work with comes from their future business courses. In order to keep the text of
the explorations relatively free of clutter, variables are defined in supplements
attached to the explorations.
Reverse and Invert 1
Reverse and Invert 2
The Swap
Solving Formulas 1
Solving Formulas 2
Balance Sheets
Simplification Moves
Solving Formulas 3
Troubleshooting
Solving Formulas 4
Mathematical Content
Pair operations with inverses, why we might want to manipulate
a formula
Compare strategies for solving 1variable equations, thinking
ahead
Introduction to reverse and invert
Reverse and invert with more steps
Introduce, justify, and use the swap
Integrating the swap with reverse and invert
Integrating the swap with reverse and invert for problems with
more steps
Compare simplification and solving moves
Explicitly use substitution into identities to simplify
expressions, begin integrating simplification moves and solving
moves
Integrate simplification moves with solving moves, more
advanced equation solving
Correcting common misconceptions that arise at this point
Solve formulas requiring reverse and invert, swaps,
simplification moves, and treating expressions as objects.
Laying the Foundation. The first exploration in this unit accomplishes two
goals: first, students pair operations with their inverses; then we investigate why
we might want to solve a formula for a specified variable, using those inverse
operations. To reach these goals, we start with the future value formula:
= (1 + )
The initial task is to figure out how much one would need to invest in order to
reach a given target (FV) within a given time (t), given an interest rate (r).
Students are given specific values for FV, t, and r and have to figure out PV. At
this point, some students solve the whole formula for PV while others substitute
the data into the formula and solve for the unknown. The next task is to take an
Excel spreadsheet with 50 clients who have specified financial targets, time
frames, and interest rates and solve the same problem for all simultaneously. This
question shows how efficient it is to solve the formula for PV and program the
result into the spreadsheet. The task aims to sell the students on the concept.
Exploration 4.2 helps students conceive of using multiple approaches to
solving the same problem, and thinking about strategy along the way. We focus
on onevariable equations. Students start by solving 2 + 4 = 10 and then
comparing to the equation:
1
2 + = 10
4
They solve the second equation in two different ways and evaluate their
approaches. Then they compare to solving the equation:
2 + 1
= 10
4
They are also directed to solve this equation using two different ways and
evaluate the approaches. Along the way, we address the role of inverse
operations and some of the choices we face. Students are encouraged to approach
the problems in the rest of the unit with a similar attitude.
Introducing Reverse and Invert. The next two explorations (4.3 and 4.4)
develop the “reverse and invert” strategy described in Part 1. Exploration 4.3 is
arguably the most important in the unit, as we introduce “reverse and invert.” We
begin with the question in Figure 2. The next question (number 2 in this
We are going to solve the equation = (1 + ) for r. In the following table, you
have been given instructions for using this formula to calculate the future value. In the
second column, write the inverse instructions in reverse order, and express each step in the
middle column algebraically in the third column. Then report the final result in the space
below the table.
Original Steps Reversed/Inverted Steps Algebra
1. Add 1 to r
2. Raise to the nth power
3. Multiply by PV
Final formula:
exploration; Fig. 3) gives students a little more to do on their own. Students are
next directed to see how reverse and invert works by concatenating instructions
and “canceling” when the instructions undo one another and are adjacent
(Question 3). Students solve = − for TR symbolically,16 and then
they respond to the question (4) in Figure 4.
Now let us solve the equation:
−
=
for Q. To do this, fill in the table below. Note that this time the “original steps” have not been
provided for you. You may add more than 3 steps if you need to!
Original Steps
1.
Reversed/Inverted Steps
1.
Algebra
1.
2.
One way to view reverse and invert is by thinking of each step you perform with an inverse
operation as appending another step to the end of the instruction list and then canceling. Add
those steps to this list for the equation in Question 3 and cancel (by crossing out) as appropriate.
− 1 for APR.
16 GP is gross profit, TR is total revenue, and COGS is cost of goods sold.
Students are asked to analyze their symbolic work in this exercise and write about
the strategic value of each step. In addition, they are given a context in which
they would want to solve this problem: programming Excel to calculate the
interest rate quoted in TruthinLending Disclosure statements (APR). Performing
this calculation assumes that the bank (the role played by the student) knows the
effective annual interest rate they want to charge (EAPR) as well as how many
times interest will compound in one year (m). Both EAPR and m are columns in
an Excel workbook. They are also asked to consider the ethical implications of
the task they are asked to perform.
What is wrong with the following attempt to solve the equation:
=
for r? Comment on both the symbolic description and the list of instructions.
=
What is wrong?
The reason for the analysis at the end of 4.4 is to keep our mathematical work
grounded in reality and connected to business. It is easy for this part of the course
to degenerate into exercises that lose their meaning. Throughout the unit, we
include questions asking students to write about their strategies, to program
solved formulas into Excel, and to return to the meaning of the formulas in
context.
Increasing Complexity. In Exploration 4.5, we introduce “the swap” as an
example of a prefabricated move, described in Part 1. The students are told in a
narrative introduction what the swap is and how it works for both subtraction and
for division. They are asked to justify both by solving:
Solve
Solve
= −
=
for B.
for C
In order to identify where the swap is and is not useful, the students are also asked
to solve both for B and to address whether swaps were necessary. Future revisions
will guide students to tie this work with the commutative property of addition and
multiplication in order to see why it works as well as why it is necessary.
We conclude 4.5 with a formula called the Sharpe ratio that is used in
finance. For stocks, the Sharpe ratio is given by:
=
−
Each variable stands for some financial quantity.17 Students are guided through
the use of the Greek letters, and then they are asked to solve for each of the given
variables. When done, students discuss which problems required using the swap
and why. Then in Exploration 4.6, students are asked to solve some formulas in
which one swap is necessary and others in which no swaps are necessary.
Exploration 4.7 guides students to integrate the swap with their use of the
reverse and invert strategy. The primary exercises in 4.7 are:18
1. Solve
=
2
(0.8)−1 for EUL.
17 The Sharpe ratio compares the “risk premium” for an investment to its risk. The variable α is the
expected growth rate of the stock, and r is the riskfree interest rate. The difference is the risk
premium – the additional reward expected for taking a risk. The denominator is the expected
volatility, i.e., the standard deviation σ of the stock. The dependent variable ϕ is the Sharpe ratio.
For a brief background on William F. Sharpe, one of the founders of the Capital Asset Pricing
Model, for whom the ratio is named, see pp. 6670 of Mandelbrot and Hudson (2004).
18 The first is the declining balance depreciation expense, as described above, using a percentage
multiplier of 2. This multiplier makes it “doubledeclining balance depreciation” in accounting.
The second formula is for straightline deprecation: SLDE is the periodic straightline depreciation
expense, CST is the initial cost, RV is the residual value at the end of the asset’s useful life, and
EUL is the estimated useful life of the asset. The straightline depreciation expense is the slope of
a linear function which returns the value of the asset as a function of time. The third formula is
called the Economic Order Quantity (EOQ). This formula tells you the optimal quantity of a good
or service to order, and it is derived by finding the vertex of a quadratic function. In this formula,
D is the quantity of the good or service demanded, H is the holding cost of keeping inventory on a
shelf, and S is the fixed cost of each order. The fourth formula is from the future value formula,
but solved for the interest rate.
2. Solve
3. Solve
=
=
−
for RV.
2( )( ) for H.
4. Solve
=
− 1 for PV.
The first requires students to see through the distractions and recognize that this
problem can be solved with a single swap. The second involves using two swaps.
The third and fourth involve a mixture of swaps and reverseandinvert.
Simplification Moves. At this point, we switch over to simplification. In
Exploration 4.8, we examine the difference between solving and simplification
moves. This difference is illustrated with balance sheets. Balance sheets balance
total assets on one side with total liabilities and total equity on the other side. The
liabilities and equity tell us the amount of the total assets claimed by creditors and
the amount claimed by the company owners. The accounting equation is:
= +
If we buy a piece of equipment for cash, we are swapping assets: cash for the
equipment. The swap is analogous to a simplification move. We have changed
the configuration of one side (which is like how we write an expression) but not
the total value of either side of the equation. However, if we borrow money to
purchase a piece of equipment, then our assets and our liabilities increase by that
amount. As a result, we have changed the values of each side, but in a consistent
way. This transaction is analogous to a solving move. After analyzing the two
transactions, students respond to the questions in Figure 6 in order to link the
theory with the algebra.
The connection between identities, substitution, and simplification moves are
made in Exploration 4.9. Students are given identities such as the definition of
adding fractions or the finite geometric series formula and asked to be explicit
about substitution when using them to rewrite expressions.19 We use the
distributive property to treat factoring out a greatest common factor as a
simplification move, which is then integrated with our solving moves when
students are asked to solve the following for t:
=
+
19 One of the design struggles in this area is balancing our desire for students to develop an “object
orientation” to equations and identities, which means they can take an equation as an “object” that
can be used or transformed, and a desire to help students understand the meaning and reason
particular identities hold. This balance is always tweaked in my annual revisions.
10. What follows is a complete solution to the equation 2( − 3) = 12. For each step,
determine whether the changes made to the equation are similar to the first or the second
transaction.
Like First or Second Transaction?
Step
= 9
(divide both sides by 2)
11. A simplification move is a step in which we change the way one or both sides of an
equation are written in ways that do not change the values of either side. A solving move
is a step in which we change both sides of an equation in ways which do change the values,
but change them in a consistent manner. Which steps in the previous question are
simplification move and which are solving moves?
12. Extend Your Thinking: What moves can you make when solving the following
problems? Are they simplification or solving moves? Are the allowed moves different for
different problems? Explain your response.
Problem 1
Simplify:
−3
( +5)( −3)
Problem 2
Solve:
( +5−)3( −3) = 5
Synthesis. Now we are ready to move to the final phase of the algebra diamond –
synthesizing solving and simplification moves. In Exploration 4.10, students are
asked to use all tools available to solve problems such as the following:20
Solve
Solve
+ ( −
) for K.
for
.
20 The first formula is Makeham’s formula for the price P of a bond. The variable K is the present
value of the redemption amount of the bond, r is the coupon rate of the bond, j is the marketbased
interest rate used to value the bond, and F is the face value. The second formula tells the relative
number N of shares of a stock to include in a portfolio that hedges risk by mixing shares of stock
with a call option on a stock. This formula assumes a binomial model for stock values: the
underlying stock price can go up to price or down to price . The variables and
represent that value of the call option if the price of the stock goes up or down respectively. Note
that this quantity should be discounted from the end of the term of the option to present value, but
is not done so in the students’ finance texbooks and so is not done so in the exploration.
The students are also challenged to solve implicit equations for a specified
variable, as in the question in Figure 7, which comes from the Sharpe ratio.21
Show that when we solve the following equation for
:
Ω
−
=
−
= Ω + (1 − Ω)
the solution can be written as follows:
In Exploration 4.11, we handle a few common errors that tend to have arisen
at least once at this point. The first issue has to do with multiplication of both
sides of an equation by a constant, but not consistently. For example, students
might start to solve an equation such as
=
+
for B by multiplying both sides by C. The error comes when they write the result
as
= +
We use written instructions in place of equations along with order of operations to
make sense of this problem. Students write the list of instructions for the original
formula, and note that appending “multiply by C” to the end of the instructions
does not “cancel” the division by C because those instructions are not adjacent
(essentially an order of operations issue). The second error concerns mixing up
the distributive and associative properties. We use our ability to carefully read
and consider identities in order to address this problem. The last error addressed
concerns dealing with rational expressions, as in the following:
Solve for m:
=
(1− )
21 The variable is the expected growth rate of a stock option, while Ω is the elasticity of the
stock option – its sensitivity to the value of the underlying stock. The other variables are as
defined for the Sharpe ratio above.
While these issues could have been addressed earlier, we felt it more valuable to
let students make these mistakes during their work, and then discuss them in this
exploration. That way, those errors are situated in their own work.
The unit concludes with some very challenging exercises, such as:22
Solve
=
+
for F
This exploration includes the most difficult of these exercises, appropriately
labeled a “challenge question,” which is to solve the BlackScholes equation:23
+ ( − ) + 21 2 2 2 =
for the interest rate r. The students are directed to treat the partial derivatives
simply as other variables, NOT as fractions.
2
Problems in the Second Course
The algebraic manipulation skills are used to solve a variety of problems in the
second course. What follows is a list of some of the most interesting problems we
address in the second course in the sequence:
1. Winners and Losers in Tax Policy: Students are asked to compare a 20%
flat tax proposal to the current progressive tax system (with 2013 rates).
They find that taxpayers with taxable incomes less than $79,400 per year
pay more under a 20% flat tax while taxpayers with incomes larger than
$79,400 per year pay less under the flat tax.
2. The Steak and Lasagna Problem: The corporate office for Hilbert’s Bar
and Grill, has decided to offer a porterhouse steak special and a vegetable
lasagna special this upcoming weekend. Students (as franchise managers)
need to figure out how many of each to prepare to sell in order to complete
22 This formula tells us how to calculate the expected rate of return ( ) on an initial public
offering (IPO) of stock where 1 is the expected dividends to be paid in the first period, 0 is the
initial price offered for the stock, F stands for the flotation costs (the cost to bankers, lawyers, etc.
required to float the IPO, as a percentage of the initial price), and g is the expected growth rate of
the stock, which is assumed to be constant. In this exercise, students are told that they are auditing
an investment bank’s IPO department and the flotation costs are missing.
23 The BlackScholes equation is a partial differential equation used to find the value V of a stock
option or other derivative financial product on an underlying asset. The variable S is the
underlying stock price and t is time. The first partial derivatives are sensitivities of the value of
the option to time and to the price of the stock. The second partial derivative is called the
“convexity” of the option (with respect to the underlying stock price). The other variables are the
riskfree interest rate (r), the dividend rate (δ) of the underlying stock, and the volatility of the
underlying stock (σ) calculated as the standard deviation.
their orders for the week. The students are given pricing and cost data
along with a budget and storage constraint. They want to place orders to
have enough to sell to maximize profits subject to these constraints.
3. Variable Interest Rates: Students adjust the compound interest formula
to account for varying interest rates and use a geometric mean to write a
formula for the “average” interest rate over a period. Some students are
bold and try (at my suggestion) to use product notation.
4. The Credit Card Problem: For a fixed principle debt (P), monthly
payment (M), and a monthly interest rate (r), students derive a formula for
how long it takes to pay off a credit card. It is assumed that the borrower is
no longer making purchases. When the students finish this problem, we
go back and determine that if we want to pay off the loan within a fixed
time frame, what our monthly payment should be. Finally, we look at the
payoff time formula and analyze and interpret the domain. The analysis
raises an important ethical question: should a bank set minimum payments
too low to pay off the debt in finite time?
5. Exponential Supply and Demand: When working with linear systems,
we solve supply and demand problems under the assumption that both
supply and demand change at a constant rate with respect to price. With
logarithms, we can solve supply and demand problems assuming supply
and demand are exponential:24
=
=
0
We also attempt a mixed problem where supply is exponential but demand
is linear, and when we find that we are unable to solve the problem, we
can see the limitations of algebraic manipulations.
6. Nonlinear Regression: Students estimate parameters for two nonlinear
functions: the national debt and production for a fictional company. Using
semilog and loglog plots, students select an appropriate functional form
to test for each data set. For the national debt, they select an exponential
model. For the production function for a fictional company, they use a
twovariable power function (called a CobbDouglass production function
in economics). Using logarithms, they transform their models into linear
functions and then run linear regressions. Correlation statistics help them
to assess their selection of a model, and they conclude by interpreting the
24 In these equations, and are quantity demanded and supplied respectively, 0 and 0 are
the quantity demanded and supplied (respectively) when the price is zero, r is the demand decay
rate, s is the supply growth rate, and p is the price.
parameters. For the national debt, they determine that the United States
federal debt grew at approximately 6.4% between 1989 and 2013. For the
fictional company, they determine that the CobbDouglass function is an
appropriate choice to model the production technology, and that the labor
productivity parameter (the exponent on the labor variable) is
approximately 0.25 and the capital productivity parameter (the exponent
on the capital investment parameter) is approximately 0.75. It follows that
the production technology exhibits constant returns to scale (doubling both
capital and labor inputs leads to doubling of the output).
The increased sophistication of these problems demonstrates how students
transition from achieving numeracy with algebraic manipulations to quantitative
reasoning with algebraic manipulations (Table 2, discussed in Part 1). These
problems represent not only increased sophistication from a quantitative
reasoning perspective, but they are also more sophisticated than problems
assigned to students in a standard intermediate algebra course (the graduation
requirement at Ferris State).
Student Performance: Preliminary Evidence
Students who took QRP have responded favorably to the approach to algebra.
One student wrote in a journal for her English class:
Equations have never come easily to me. It was always here is the formula now plug
numbers in and hope for the best. I never understood why and [the fry efficiency
exploration] helped me with that. I realized the reasoning behind equations, what it
meant and how people put them together. … I found out there is logical thinking behind
formulas and it isn’t random variables thrown together.
That is exactly the point, of course. We want students to see meaning and
purpose in algebraic communication.
Student performance on problems provides some evidence of success. At the
beginning and end of this module, we asked students in QRP in the fall of 2016 to
anonymously complete the following four problems:
1. Solve for Q:
2. Solve for T:
3. Solve for W:
4. Solve for K:
− 12
2
2
= − (
)
The reader may recognize these from the description of the levels of difficulty in
Part 1. In addition to asking QRP students to solve these problems, we also asked
students at Ferris State completing beginning algebra, intermediate algebra,
precalculus, and calculus 1 to do the same (anonymously). The goal is to match
QRP student performance to an appropriate “grade level” by comparing the
percentage who solved each problem in QRP to their counterparts in the four
standard courses. Note that none of the QRP students were able to solve any of
the problems prior to working through the module. We did not collect “before”
data for the students in the more traditional courses, as the purpose of those data
was to establish “grade level” performance.
Because we are comparing proportions, we use a ztest for significance when
comparing two proportions and a 2 test when comparing more than two
proportions. For some comparisons, the sample sizes are fairly small which may
contribute to the lack of evidence of significance at this point. The small sample
sizes in those cases explain why I consider these data to be “preliminary”
evidence of success. Nevertheless, we feel that the data are powerful, and in
sharing them, it is important to assess statistical significance.
Placement into the QRP and the four comparison courses, as well as sample
sizes, are in Table 6. To start, one question we can consider is whether student
performance indicates that the difficulty levels of the problems are as we thought.
Figure 8 displays the percentage of all 757 students who correctly solved each
problem. Using a 2 test, we
found p < 0.0001. Therefore,
there does appear to be a Correct Solutions: All
statistically significant difference Students
in the difficulty levels across the 30%
four problems. However, when
we considered pairwise problem 20%
comparisons, a twosample ztest
gave us p = 0.2955 when 10%
comparing Problems 1 and 2, 0%
with p < 0.00001 in all other Problem 1 Problem 2 Problem 3 Problem 4
cases. This result suggests that
there may be more subtle eFaigchurpero8b.lPemer.centage of students who correctly solved
differences in the first two
difficulty levels. More specifically, we suspect that, for some students, the
number of operations and variables in the first problem outweighed the challenge
of solving for a denominator in the second, whereas for some other students the
opposite was the case.
More importantly, we can use the data to determine how students who took
QRP compare to students in the standard four courses (Figs. 9‒12 and Tables 7‒
10). The data suggest that the performance by QRP students is around the level of
precalculus. The difference between the percentage of QRP students successfully
completing each problem and the corresponding percentage of their peers in
beginning algebra, as well as from those in the more advanced intermediate
algebra course, is practically and statistically significant. The highest percentage
of students successfully completing each problem came from calculus
(thankfully!). For all but one of the problems, the statistical significance between
QRP and calculus is questionable. Given the small sample sizes for calculus and
QRP, we cannot form a definitive conclusion.
Correct Solutions: Problem 1
70%
60%
50%
40%
30%
20%
10%
0%
Beginning Intermediate Precalculus Calculus 1
Algebra Algebra
QRP
Correct Solutions: Problem 2
80%
70%
60%
50%
40%
30%
20%
10%
0%
70%
60%
50%
40%
30%
20%
10%
0%
Beginning
Calculus 1
QRP
Correct Solutions: Problem 3
Calculus 1
QRP
Correct Solutions: Problem 4
60%
50%
40%
30%
20%
10%
0%
Calculus 1
QRP
The closest comparison appears to be with precalculus students. The
magnitude of the pvalues for the first three problems suggests that there may not
be a statistically significant difference in these cases, although the small sample
size for the QRP students indicates a need for more data. For the fourth problem,
QRP students perform better than precalculus students do, and the difference is
significant at the .05 level but not the .01 level. This result suggests that a larger
sample may yield a statistically significant result.
I should point out that students in QRP were tested before and after a module
that was specifically designed to improve algebraic manipulation skills. The
instrument itself included two problems that could be solved using “the swap.”
Further work should involve developing an instrument that is open to additional
prefabricated moves. Finally, note that the QRP students completed this
instrument immediately upon concluding the module. A further longitudinal
study might yield different results, and could also assess whether students grow
from being numerate with algebra to being quantitatively literate and able to
quantitatively reason with algebra, as described in Part 1.
Despite these limitations as well as some small sample sizes, the data show
promise. I believe part of what we are seeing is the impact of sensemaking.
Guiding students to identify appropriate steps themselves within the framework of
treating algebraic expressions as instructions, appears to lead students to
purposefully use solving and simplification moves to unravel the formulas.
Another part of what we are seeing comes from contextualization. We create a
need for formula manipulation within realistic contexts, aided and abetted by
Excel. The relative importance of these factors is unknown at this time – our
preliminary success comes from a “cocktail of treatments” so to speak. In future
studies, we might conduct interviews and collect qualitative data about the
students’ thinking process and their motivation. Nevertheless, our preliminary
data suggest that this combination of the “instructions perspective” and
contextualization has the potential to lead to improved algebraic reasoning.
Concluding Remarks and Further Questions
Numeracy, quantitative literacy, and quantitative reasoning are not “nonalgebra.”
These terms provide a framework for assessing depth of understanding of
knowledge domains from mathematics and statistics. This framework fits well
with algebra, at least within my perspective of what we mean by “algebra” and
“algebraic manipulations.” This perspective yields a “quantitative reasoning”
approach to algebra that can be operationalized and taught. By applying this
approach using inquirybased materials motivated by contextualized problems,
students construct an understanding of algebra and develop the skills within the
framework of the algebra diamond.
Some preliminary data show the potential this approach has to be effective,
notwithstanding limitations. However, using the analysis of Karaali et al. (2016),
we note that we are measuring only numeracy with respect to algebra. There are
many further questions and directions that can be explored. Can we guide
students to use generalized “prefabricated” solving moves besides the swap?
What would happen if we treated “negation” and “reciprocation” as operations
which invert themselves? Would it be worth the time to guide students to replace
inverse operations with inverse elements?
Further down the road, we can ask whether students who complete the second
course have achieved my standard for quantitative reasoning with algebra. We
can ask if their growth persists into their studies after completing their math
courses. We can also look for the impact on affective characteristics, such as
math anxiety and beliefs about math. For example, I have some preliminary data
that suggest students who complete QRP experience a significantly larger
reduction in math anxiety than those who complete a projectbased liberal arts
math course that eschews algebra and algebraic manipulations (while both sets of
students experience a significantly larger reduction than those who take
intermediate algebra). My hypothesis is that the students in QRP recognize the
algebra we do in class as “math,” while students in the liberal arts math course
think something along the lines of “well, that course was nice, but in a real math
course…” I won’t know without conducting interviews.
There is also a question of whether this course or its materials can be brought
to scale. So far at Ferris State, four faculty members have taught the course
(including me, the course’s creator). They each taught the course under different
circumstances and have found the experience rewarding. The data presented in
this paper aggregate the results from three sections of QRP taught by two
different instructors, including myself. Additionally, it is appropriate to ask if the
course can be adopted for students in other occupational majors, such as health
professions. In the near future, I have two plans for scaling and expansion. One
plan includes adding nursing and social work students to the audience for the
course. This further work involves collaboration with nursing and social work
faculty, and it is supported by a grant from the National Science Foundation. The
other involves using a faculty learning community to extend the number of
instructors teaching the course.
I believe that with QRP, I have stumbled into something with great potential.
This paper can serve as an “existence theorem” illustrating that algebra can be
addressed within the framework of quantitative reasoning. The students’
performance suggests that quantitative reasoning is a powerful framework for
learning algebraic manipulations. The problems students can handle in the second
course of the semester serve to underline this promise. There remain a lot of
questions to ask, and fertile ground for future research. I invite the readers to join
me in this journey.
Acknowledgments
I would like to thank all of the instructors who delivered the algebra instrument to
their students. I also thank Anil Venkatesh, Erin Militzer, and Harvey Hannah for
their willingness to teach QRP. In addition, Kirk Weller has served as a valuable
resource and sounding board through course development and assessment studies.
I would also like to thank the referees who reviewed this paper for their helpful
suggestions. While I wish the final draft was shorter than the initial draft, it seems
that whenever I revise papers I am confronting Hyrdra: for every sentence
deleted, two more take its place. In addition, I thank Mike May from Saint Louis
University, whose Excelbased business calculus course inspired the use of Excel
in QRP. Finally, I would like to thank Len Vacher and Eric Gaze, each of whom
recommended that I submit this paper to Numeracy.
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