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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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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Math Applications — Taking it Personally

Contextualized math is a current theme in our profession; some courses are taught strictly working from context — the story is the thing, and only the mathematics that relates to that story is developed.  Other courses emphasize context, while many of us take a moderated approach in which we blend context and abstraction.  Whatever the mix, these contexts are experienced as ‘applications’ or word problems by our students.  Do applications (or context) present issues of equity in our developmental math classes?

I would like to tell you about what two of my students (beginning algebra) are experiencing this semester.  First, a student to be called Mary.

Mary: (looking at a typical ‘distance’  problem about two cars)  I don’t know how to write the algebra for this, but I can figure out the answer.
Instructor: Okay, so tell me more about that.  How do you figure it out?
Mary: Well, the problem says that one car is going 10 miles per hour faster, so I put myself in that situation; I know that the speed limit is 70 miles per hour, so that must be the faster car.  The other car must have been going 60.
Instructor: I see.  What part of the problem told you that the cars were on a highway with a speed limit of 70?
Mary: The problem did not say that, but the only way I can understand the problem is to put myself in to it.

The second student will be called John (whose native language is Arabic).

John: (looking at a problem about a tree and a flag pole dealing with their heights)  This problem is really hard.
Instructor: What makes it hard?
John: Everything in the problem … I need to translate it into my language; it does not make sense to me.
Instructor: Are you talking about the individual words?
John: Yes, yes … they are confusing.

The prognosis for Mary is not as good as the prognosis for John.  They are both taking the applications personally; the difference is that Mary thinks that she has to see herself in the problem for it to make sense, while John thinks that he will understand the problem once he knows all of the words.

This experience made me think of some research I saw a few years ago dealing with how word problems in mathematics might raise issues of equity.  The research suggested that students from a ‘lower class’ (this was British research) get distracted by the details of the applications as they relate to their personal life.  My student, Mary, was doing exactly that.  Her learning skills, and her life experiences, provided a limited view of applications; some problems dealt with objects or situations with which she had no experience, and she did not know what to do … other problems activated related but not worthwhile information (like the car problem).   Clearly, we will need to work together (Mary, the class, and me) to help broaden the view and provide more resources.

Taking an application personally can create difficulties in forming a solution strategy; taking it personally highlights information (which might be trivial) and causes us to possibly ignore other information critical to a solution.  This situation deals with perception and motivation.  For those of us who are using high-context classrooms, I wonder if you are finding that the approach is equally accessible to all of your students.

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Math Applications … Magic of “Is”

What price are we willing to pay for ‘correct answers’?  What gains (benefits) should students expect for dealing with applications in a math class?

In our beginning algebra class this week, we spent much of our time on applications.  Many of these were the typical puzzle problems involving tickets and cars, integers and angles.  As is normal for this course, students really wanted some magic — a rule that would help them get the correct answer for all of the problems.  Some of the students remembered some magic from a prior math class; one piece of magic was the word ‘is’ … the other piece of magic was a triangle (for mixtures).

We often provide rules (whether perfect or not) that are meant to help students get more correct answers for applications (broadly stated as word problems involving a context).  We tell students that “of” means multiply, and that “is” means equal; the prototype for both rules is the “a is n% of b” template (a worthless model, as normally taught).  Students who have experienced this ‘correct answer’ driven course encounter many problems when faced with a narrative about an application, where ‘of’ is the normal preposition and ‘is’ is the normal verb connecting phrases.  We train our students to surface-process language for the sake of correct answers, and wonder why students continue to have problems with applications.

One of the most challenging problems we did this week was this simply-stated problem:

A store claims that they markup books by 30%, and the selling price for one book is $79.95. Find the cost of the book to the store (before the markup was added).

Every student in this particular class was a graduate of our pre-algebra course, where this same problem was done as part of a longer chapter on percents and applications.  Every student in this class wanted to either multiply by 30% or divide by 30%; a few students thought that there was a second step where they needed to add or subtract this result.

Quite a few of the students could do this problem:

A store sells a book that has a cost of $61.50, and they have a markup on books of 30%. Find the selling price.

Their success on this arithmetic problem was not based on understanding the words any better (the words are the same).  Their success was based on the ‘magic’ rules we had given them that happen to work: multiply by the percent, add or subtract if needed.

The whole point of experiencing applications such as these is to build up the student’s mathematical reasoning.  There might be magic in the world, but magic is not reasoning.  Correct answers based on locally-working magic is worse than wrong answers based on weak reasoning.   If our courses include applications, keep the magic of “is” out of the course … and all other magic. 

 
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