Algebra, the Quadratic Formula, and Life

Like most of us, I am still teaching a traditional developmental algebra course — two of them, in fact (beginning algebra, intermediate algebra).  This traditional content includes standard topics, and one of them has been bothering me for some time.  Can we drop the quadratic formula from our courses?

First, let’s come up with a list of good reasons to study the quadratic formula.  Well, did you think of any?  The most common rationale given is that “the quadratic formula solves ANY quadratic equation”.  That is partially true; the formula CAN solve any equation that is strictly quadratic (but not equations that are quadratic in form).  Let’s ignore that shortcoming, as perhaps it is not a significant gap.  The importance of the quadratic formula, then, depends on how important it is to solve a strictly quadratic equation.  Is it the only way?  Of course not — completing the square can also solve any strictly quadratic equation … often with less computational effort.  We also can use numeric and graphical methods to solve any quadratic equation.   Another common rationale for the ‘QF” (as it is sometimes known) as it facilitates the use of the discriminant; whether the discriminant is worth the bandwidth depends on how we use it, and how the study of the discriminant contributes to the mathematical reasoning of our students.  Some people use the QF to determine linear factors of quadratic expressions, which fits in to the ‘correct answer’ world view; I doubt if using the formula to factor expressions contributes to an understanding of equivalence.  [However, I have to admit that our normal instruction of factoring is not really designed to produce understanding of equivalence.]

How about good reasons to NOT study the quadratic formula?  Well, did you think of any?  Quite a few of my students dislike the formula because they realize how likely they are to make a minor arithmetic mistake which results in catastrophic failure to solve the equation.  Some of these students have a strong preference for completing the square — because it provides a logical sequence of steps that avoids many mistakes.  We also have a mythology among our students that says success in mathematics depends on the mastery of formulas to generate the correct answers required in a class.  Few of us concur with the importance of ‘correct answers’ in that myth, but many of us contribute to the myth by placing an emphasis on the quadratic formula.  I would say that the use of the quadratic formula to solve an equation detracts from the mathematical reasoning that I am trying to develop in my students.

Of course, the ‘elephant in the room’ with us is the role of quadratic equations and expressions in general.  Why are they important?  We could spend several blog posts on that topic, and we might go there someday.  For today, here is a brief summary:  the quadratic equations are included to foreshadow some authentic uses in STEM courses later, so we include some puzzle problems that result in quadratics in the developmental course (rectangles of a certain area, projectile under the influence of gravity; we also use quadratic equations as a field test of other algebraic skills (factoring, radicals, complex numbers, etc).  Very few processes (either in nature or in society) are essentially quadratic; the most common quadratic equations in valuable applications come from modeling data (such as fuel efficiency vs speed, profit vs production, etc). 

Very few of the applications leading to quadratic equations have a value in helping our students become more sophisticated in mathematical reasoning, nor in problem solving in general.  The solving of these problems is an exercise.  Therefore, this exercise should develop something of value in our students … and this does not mean ‘correct answers’.    Many applications are solvable by using square roots (like x² = 18), and that method can be connected to a series of related knowledge.  If the problems involve a full-quadratic, numeric & graphical methods provide solutions to most with connections we can make to other knowledge.  Resorting to the quadratic formula bypasses connections and understanding the process, and the QF stands isolated from other knowledge (for almost all of us).

How about a reality check:  How many of us “reach for the formula” to solve a quadratic equation arising from a situation or problem that is worth solving?  These problems often involve non-integer coefficients.  We are likely to reach for the QF primarily when the solutions are complex numbers, where numeric and graphical methods are less accessible.

Unless we teach the Quadratic Formula in a connected fashion, richly connected to basic concepts of mathematics, I think we do more harm than good.  Without those connections, the formula reinforces the myth of right answers.  Mathematics is important in life; the quadratic formula has few contributions to make.

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What is Our Plan?

In the classic problem solving methodology, the intense effort is placed in two early stages — understand the problem, and make a plan.  In the case of mathematics (especially developmental mathematics), we have seen much hand-wringing and gnashing of teeth, often accompanied by saber-rattling, about ‘the problem’.  However, a problem can only be defined by comparing the current condition to the desired condition.  Looking at data is a first step, but can often lead to short-sighted efforts that do not solve any significant parts of the problem.

Here is one overview of a plan for mathematics in community colleges (focusing on developmental mathematics, though not restricted to that):

• All math courses must provide good mathematics (appropriate and powerful concepts to deal with quantitative situations).
• All math courses must prepare students for mathematical needs that they will encounter in college.
• Community college mathematics is not a repeat of school mathematics.
• Community college mathematics is compatible with, and supportive of, university mathematics.
• Reasoning and problem solving are central goals of mathematics as part of a general education.
• Remediation is needed for some students, ideally limited to one course or a fast-track experience for most of those students.
• Any student might be inspired to higher goals, and many are capable of additional mathematics in a reasonable amount of time.

If a “solution”, whether modules or online homework or emporium model, only deals with the patterns of the data, then the solution will not solve anything important.  In some cases, the ‘pass rates’ might rise temporarily or even long-term; however, there is still likely to exist a substantial gap between a larger plan for mathematics and what is actually delivered to students.  If the traditional mathematics does not contribute to a larger plan (which is my view), then a solution plan involves much more than the delivery system and much more than course organization.

In the case of developmental mathematics, we have a historical artifact which is based on a premise that we need to provide the same mathematics that students should have learned in high school.  Such an approach is arbitrary, unrelated to mathematical needs, and dooms our courses … and dooms our students … at the system level.  Having a sequence of 4 courses in developmental mathematics guarantees that less than 20% of the students will reach college work, based on an unreasonably high 80% pass rate and 80% retention rate.  The response, based on ‘the data’, is to get students to their exit point in this ‘school mathematics’ as quickly as possible (modules); is our plan for mathematics that students should be shoved off the train as soon as possible … or do we want to have an opportunity to inspire students?

The emerging models — AMATYC New Life, Carnegie Pathways, and Dana Center Mathways — are based on a larger plan.  However, many of us are looking at them as responses to ‘the data’.  For these models to work well, the faculty and colleges involved need to have a deeper understanding of a plan for mathematics.   Hopefully, you will see much in the plan outlined above that you can agree with.  One of our basic problems is that policy makers do not have this larger plan for mathematics in mind; they, naturally, focus on the data.  We, and our professional organizations, need to articulate a larger plan so that we can better serve our students.

One of my colleagues said, back in 2008, that pass rates are the least of our problems in mathematics.  I agree.  We need to have a plan for mathematics, and build new curricula to support that plan.

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MLCS, Quantway – History and Comparison

In this post, I would like to clarify how Quantway™ and MLCS (Mathematical Literacy for College Students) compare.  Part of this will involve a brief history of these courses; the rest will be a focus on the practical differences — keeping in mind the considerable overlap that exists.

The starting point for this work was fall of 2006, when AMATYC released the standards “Beyond Crossroads” (http://beyondcrossroads.amatyc.org/).  At the heart of the discussions on appropriate content, the standards emphasized quantitative literacy in all math courses.  The release of the standards occurred during the AMATYC conference that year; at that conference, members of the Developmental Mathematics Committee had a brief discussion about models for developmental mathematics … essentially saying that it would be nice to have an actual model to guide our work, instead of a curriculum constrained by history.  Over the following two years, conversations took place about the problems in developmental mathematics and creating a plan.

By 2008, these conversations within AMATYC led to the start of the New Life Project by the Developmental Mathematics Committee (DMC); we formed some work teams which developed a collection of learning outcomes that followed from professional work outside of AMATYC (Numeracy Network, MAA, etc) as well as within AMATYC (Beyond Crossroads).    To help our work, we created an online community in early 2009 — the wiki at http://dm-live.wikispaces.com, and invited professionals to join the community. 

During this same period, foundations were becoming more interested in the needs within developmental mathematics.  The resulting opportunity for collaboration (especially with the Gates Foundation) led to a Seattle meeting in July of 2009; these people (sometimes called the Seattle 15) created the first draft of a curricular model — the first model created for developmental mathematics, designed for broad implementation.  This model identified two courses; the first course was originally called ‘the blue box’ (because that was the marker color used that day), and the second course was called ‘transitions’ (because we thought ‘the green box’ might not work too well).  As the model was developed, it became clear that we needed to deal with other factors — especially professional development.

This timeframe coincided with the Carnegie Foundation for the Advancement of Teaching starting their pathways work.  Members of the New Life Project were included in all of the original planning for the pathways work, and the initial learning outcomes were those of the ‘blue box’ course.    As Carnegie worked with their curriculum partner (Dana Center, University of Texas – Austin), these learning outcomes were vetted by professional organizations  and kept synchronized with the New Life work.   (Most of these same learning outcomes exist in the Statway courses.)  The Pathways work included a deliberate system for professional development, called the Networked Improvement Community (NIC).  Although the NIC was developed without direct input from New Life members, the design of the NIC dealt with the same professional issues that New Life identified.

Originally, the Pathways course was called “Mathway” and the New Life course with the same content was called “Foundations of Mathematical Literacy”.  Each of these names had problems.  By the end of 2010, the current names were identified — Quantway and Mathematical Literacy for College Students (MLCS).    That year (2010) was the first year that faculty at particular colleges became interested in beginning the process to implement the new course; most of the early interest was in MLCS, because the online community could communicate at that time … the Carnegie work with colleges came a little later.  Several colleges that were among the first to be interested in MLCS decided to become part of the Quantway network.

When Quantway colleges developed their courses, they sometimes named their course “MLCS” — the content of Quantway and the New Life MLCS are essentially the same.  Indeed, there are high levels of agreement between Quantway and MLCS in content and professional areas.  Some colleges outside of the Quantway NIC say that they are implementing Quantway — this is not true; implementing Quantway means that your college has been accepted formally into the NIC.  Outside of “the NIC”, colleges are implementing the New Life MLCS.

That is the major difference between MLCS and Quantway: Quantway involves a formal network (NIC), with commonality of implementation; MLCS (New Life) involves a local implementation of a model course adapted to local needs, with an informal network.  The New Life project operates as a subcommittee of the DMC, and we continue to develop resources to support faculty.

A related difference lies in the materials.  Quantway colleges all use the same materials (currently Quantway version 2.0), which includes an online system and common assessment items.  MLCS (New Life) faculty use either commercial texts or locally written materials; in some cases, the locally written materials will be developed by publishers into commercial texts.  The Quantway materials are currently about a year ahead of MLCS materials — MLCS materials are at the pilot or class test phase (1.0) while Quantway is at version 2.0. 

The second major difference lies in the curricular purpose for the course.  Quantway is intended to be the prerequisite to a quantitative reasoning course (aka quantitative literacy); this is how the name was chosen — and Quantway 2 (the second course) is currently being developed by Carnegie.  The New Life MLCS course can also be used for this purpose; however, the MLCS course is seen as playing a larger curricular role — MLCS can be the prerequisite to an introductory statistics course, other quantitative reasoning courses, and the Transitions course.  If colleges implement both MLCS and Transitions, they can completely replace their developmental algebra courses.  In other words, Quantway is designed to serve very specific groups of students; New Life MLCS is designed to be the basis for fundamental change.

The other difference lies in the practical issues of implementation — Quantway colleges must use the implementation process of the NIC, while MLCS faculty can do their own or use the resources of the New Life project.  The New Life project focuses on helping faculty and departments meet local needs with flexibility; the Quantway process emphasizes the NIC with limited options for local adaptations.  Again, both models incorporate needs outside of the content; professional development is critical in both.

All other differences are matters of aesthetics and minor details.  If you remember that Quantway and MLCS share a common source, and that the differences lie in networks and implementation, you will have a fairly accurate view.  The two labels are not equivalent, because the Quantway label includes the NIC and commonality of implementation; New Life MLCS includes the long-term reform of the curriculum (combined with Transitions).

I invite you into this process of bringing new life to the developmental mathematics programs that serve the needs of our students.  We can escape the ‘black box’ of history, and enjoy a ‘blue box’ and a ‘green box’ — MLCS and Transitions.  Your department can begin this work.  Your college might choose to apply to become part of Quantway.  Take that first step on the road to better mathematics for your students.

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PEMDAS and other lies :)

We use ‘correct answers’ as a visible indication of knowledge.  If the learning environment focuses on homework systems, correct answers may be the only measure used.  However, even when we ‘look at the work’, we may confuse following a procedure with knowing what to do.

PEMDAS may be the most commonly used tool in the teaching of mathematics experienced by our students.  I have seen PEMDAS written on work papers and notebooks; I have heard students say PEMDAS when explaining how to ‘do a problem’ … and I’ve heard instructors say that PEMDAS describes what to do with an expression.

The problem is that PEMDAS is a lie.  PEMDAS only provides a memory tool (a mnemonic) for steps that might apply to some expressions in some situations.  Previously, I have written about the issues with the “P” (parentheses) component of this tool (see https://www.devmathrevival.net/?p=301).  Today, I am thinking about some of the ways in which PEMDAS is false or incomplete.

Take a simple expression like -4².  PEMDAS does not give any interpretation of this expression.  The issue here is that the memory aid only deals with exponents and the 4 binary operations; the negation (opposite) involved here is outside of the rule.  If we established mathematical truth based on an agreement among students passing a course, the truth would be at risk on this expression — whether “16” or “-16” would win a majority would vary by semester.

PEMDAS is incomplete about operations in general, such as the negation above … or absolute value.  Given the visual similarity with parentheses, most students see that the ‘inside’ of an absolute value is simplified first.  However, what to do with an expression like  3|x – 2|?  Is there a choice to distribute?  As we know, and students are confused about, the order of operations provides one possible procedure … properties of numbers and expressions completes the story, and these properties are more important in mathematics.  Getting the correct answer to 8 + 5(2) in a pre-algebra course has nothing to do with being ready to succeed in algebra, or math in general.  Basic expressions like 8 + 5x are a challenge for many students, partially due to how strong the PEMDAS link is.

Another example:  what does PEMDAS tell us about mixed numbers?  This is a special case of the ‘parentheses problem’, where there is no symbol of grouping.  Fractions, in general, are an area of weakness.  We tell students that “you need a common denominator” or “cross multiply” — both of which appear to violate PEMDAS (we would divide left to right).  Properties are the important thing here as well; adding requires similar objects.  We focus so much on correct answers and perhaps ‘correct steps’ that we miss opportunities to address the mathematics behind the visible work.

The meaning of an expression with mixed operations is based on the priority of each operation; mathematically, the level of abstraction of an operation determines the priority.  Multiplying is abstracted from the concept of repeated adding, so multiplying carries a higher priority; exponentiation is abstracted from the concept of repeated multiplying, and has a higher priority.  Lowest abstractions are the basic concepts — add, subtract, negation.  For those of you involved with programming, this approach should sound familiar — computing environments are based on a detailed list of these levels of abstractions.  In mathematics, our world is defined by properties which provide necessary choices for types of expressions where equivalent forms can be created without using the prioritization.

The big lie in PEMDAS is that those 6 words say something important about mathematics.  Those 6 words do not say anything important about mathematics, only about an oversimplification that produces some correct answers to some expressions without understanding the mathematics.  Properties and relationships are the important building blocks of mathematics; a student starting from PEMDAS has to unlearn that material before understanding mathematics.   If our goal is to have students compute correct answers for any expression, then we would never use PEMDAS — it is woefully incomplete, and we would need the prioritization list like a computer program uses.  If our goal is to have students understand mathematics, we would deal with the concepts that determine the order along with the properties that provide choices; a focus would be on the correct reading and interpretation of expressions.

Do your students a favor; avoid using PEMDAS.  Use mathematics instead.

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