MOVEMENT 3
The solution which I am urging, is to eradicate the fatal disconnection of subjects which kills the vitality of our modern curriculum. There is only one subject-matter for education, and that is Life in all its manifestations.
—Alfred North Whitehead (1861–1947), The Aims of Education and Other Essays
6
By whatever means it is accomplished, the prime business of a play is to arouse the passions of its audience so that by the route of passion may be opened up new relationships between a man and men, and between men and Man. Drama is akin to the other inventions of man in that it ought to help us to know more, and not merely to spend our feelings.
—Arthur Miller (1915–2005), Introduction to Collected Plays
To acquire an intellectual advantage at great cost, if it can be attained more cheaply is unnatural…
—Carl von Clausewitz (1780–1831), Clausewitz and the State
On Monday, January 6, 1930, the great American philosopher and education theorist John Dewey was invited to give a series of public lectures on philosophy and psychology at Harvard University.1 He accepted on January 13, and one year later, he gave a sequence of ten talks so extraordinary that their content was later collected in a book called Art as Experience (1934).2
Scattered among the book’s sometimes dense passages are conceptual gems of such dimension, creativity, and zing that they continue to rouse the imagination:
Every art communicates because it expresses.… For communication is not announcing things.… Communication is the process of creating participation, of making common what had been isolated and singular.3
A primary task is thus imposed upon one who undertakes to write upon the philosophy of the fine arts. This task is to restore continuity between the refined and intensified forms of experience that are works of art and the everyday events, doings, and sufferings that are universally recognized to constitute experience.4
We have an experience when the material experienced runs its course to fulfillment.… A piece of work is finished in a way that is satisfactory… is so rounded out that its close is a consummation and not a cessation.… The experience itself has a satisfying emotional quality because it possesses internal integration and fulfillment reached through ordered and organized movement.5
Dewey advances the view that many aspects of everyday life have the potential to be aesthetic experiences, and that they should be recognized as having this capacity. He felt that artistic activity is not exclusively the domain of the fine arts, where it is intentionally created to be experienced—as enlightenment captured or tamed—often in museums or galleries, but rather that the aesthetic experience can also happen as events unfold and unite in wild, free-flowing forms.
The artistic experience comprises a continuum between its fixed forms on the one hand and the high points of everyday life on the other, where actions and ideas converge together in unison: the smooth execution of a well-crafted, victorious game plan or campaign strategy, the successful, mature handling of a stressful task, the beautiful rendition of a song at graduation, the enjoyment of a long-desired vacation, or even a crossword puzzle or sudoku grid completed in a creative way after some struggle. Such peak experiences also include the epiphanies that occur in education: namely, moments of insight where new concepts and old memories converge together in students so that they suddenly see or understand something in a strikingly fresh or very clear way.
Dewey thought that instructors should leverage such moments in their teaching and deliberately engineer scenarios to assist students in experiencing the excitement and satisfaction of substantive understanding along with the joy of insight, which has been described as being “a sense of involvement and awe, the elated state of mind that you achieve when you have grasped some essential point; it is akin to what you feel on top of a mountain after a hard climb or when you hear a great work of music.”6
To date Dewey is still recognized by many as the preeminent education thinker of American origin.
CONCEPTUAL FUELS
The great acting teacher Stella Adler shared some of Dewey’s sentiments about unity and continuity in her own field, telling her students, “Your curse is that you have chosen a form that requires endless study.… It means you have to read, you have to observe, you have to think, so that when you turn your imagination on, it has the fuel to do its job.”7
Widely considered to be one of the twentieth century’s leading teachers of drama, Adler believed that imagination and research rather than personal memories or emotional recall should inform an actor’s craft, and that much of what an actor learns, observes, and thinks about can serve as fuel for their performances. It is a powerful statement because she is not suggesting passivity in having this happen either, but rather appears to advocate that actors actively and aggressively employ their individual point of view—their observations, thoughts, and imagination—as conceptual fuel for acting.
Quotations serve as fuels for the imagination, too. They are very popular, and many people are known to inventory collections of them for study and use as well. Why do they do this? Some do it for fun, but I dare say that a large portion also do it for the down-the-road purpose of later using the quotations to motivate, inspire, or enlighten others (or their future selves) for purposes or situations that may have been unknown to them at the time that they first decided to save the quote.
The thing that makes quotes so appealing to us is due in part to the fact that situations in life rhyme and sometimes an apt demonstration of a relationship or a phenomenon in another area (or at a different time) may be encapsulated in a quotation. The statement may provide a spot-on conceptual perspective or game-changing orientation to a more abstract, unpredictable, or confusing situation. Quotations communicate, across the ages, other people’s singular moments and insights—serendipitous times where they have been gifted with the ability and vision to snatch something transcendent and eternal from a profound, imaginative, or confluent instant in time or thought.
When such a statement powerfully resonates with a person, it is almost like a window in time opens up for them to a wider world of shared experiences, allowing them to momentarily glimpse, “in the palm of their hand,” some portion of the past, present, and future all at once.8 These junctures become like individual little packets of Dewey-type experiences in and of themselves. And their preservation is a spectacular thing to have available at our fingertips. People’s use of quotations as fuel for their own interpersonal interactions and private inspirations approximates what Adler is telling her acting students to do in employing the results of their “endless study” to fuel their various performances. It also approximates the conceptual fuel that algebra can supply in providing greater insight into certain types of quantitative situations.
WHAT TO DO ABOUT WORD PROBLEMS
Dewey’s and Adler’s thoughts are highly relevant to the case for algebra in general, and algebraic word problems in particular.
How best to incorporate such problems into the classroom is one of the central issues in the teaching of algebra. But it is also one of the most contentious. Most educators agree that word problems should be included in any such teaching, but they disagree mightily on how much they should be used, where they should be introduced, how they should be integrated, and what types of problems should be used.
One common complaint is that most word problems are contrived and artificial, mostly about irrelevant things or fanciful situations. Who cares anyway about silly rope lengths, perimeters of hypothetical geometric figures, the measures of made-up angles, or the number of hamburger meals in an imagined business? Is anyone ever really going to ask us to pay for something with so many quarters and dimes, exclusively?
Many critics claim that such trifles are among the reasons that so many students get bored with the subject and wind up ultimately disliking algebra. If students could only be exposed to the real uses of algebra, the thinking goes—if they could see algebra applied to realistic situations in finance, physics, accounting, biology, and computers—then they would learn to appreciate its power and relevance to the world at large.
Many experts agree with this assessment on the state of word problems in math education.
But is this really enough on its own: to simply make word problems more about realistic situations and less contrived, as in the last chapter, and then magic will happen, with students turning on to algebra and appreciating it? Noted University of Virginia cognitive psychologist Daniel Willingham doesn’t think the solution is so straightforward.
In response to the 2013 New York Times editorial “Who Says Math Has to Be Boring?”, which reiterates that real-world application is the key, Willingham writes: “So the proffered solution is real-world application. But I think a worse problem is not understanding how math works, being asked to execute algorithms with no understanding of what is really happening.”9 He goes on to mention in his book, Why Don’t Students Like School, that students can be easily bored by information, even if the material is relevant to them.10 Most teachers would probably agree with this observation.
Willingham cites his own experience at how even he can be turned off by presentations on topics critically related to his own interests and area of expertise, especially if poorly presented.11 Many of us have probably had a similar experience of being excited and highly motivated to attend an upcoming talk, read a new book in our area of interest, or watch a film adaptation of a book series we love only to be disappointed, bored, or downright frustrated by the nature of its presentation.
All of this suggests that simply shifting the focus to real-world and “relevant” problems won’t be enough to make magic happen on its own. More is required—something akin, perhaps, to sharing with students more of the types of fulfilling experiences and unity of ideas that Dewey and Adler talk about.
But where does that leave the role of word problems in math education? Are they obsolete architecture left over from a pre-digital world? Or are they an essential tool for appreciating algebra’s wide-reaching usefulness? I believe that there is a place in the teaching of algebra for both contrived problems and the problems drawn from real-world applications: in the workplace, from newsworthy events, from science, or in other areas of everyday life. But in both cases, the manner of their presentation and coordination along with an awareness of the needs of their intended audience are absolutely essential to the success of the endeavor.
CLASSROOM WORD PROBLEMS
The distinction between contrived word problems and more realistic problems can be blurry at times. For instance, the break-even point example from Chapter 4 involves a scenario from business (hence real-world in appearance), yet the numbers we used were engineered such that the break-even point would work out to be a nice whole number. Moreover, they were chosen without the market research one should normally do when starting a business. All of this means that this particular problem has a real-world aspect combined with a manufactured portion.
In recognition of this, we will make a slight adjustment in our classifications. Now, in addition to the truly manufactured situations such as the number of days and age problem (or rope length problems), we will join together those real-world-flavored applications and intentionally sculpted scenarios not likely to actually occur—and place them under a new category that we will call “word problems for the algebraic classroom” or, more simply, classroom word problems.
For now, we will reserve the category of real-world applications for those quantitative problems that some person might naturally encounter outside of an algebra class or recreational math book. There will obviously still be much overlap between these two redefined categories, but these are the rough subdivisions that will be meant in the discussion that follows. I share Willingham’s view that we should teach for understanding, a deeper type of understanding that involves more than simply knowing how to execute algorithms. This understanding certainly includes knowing how to process algorithms, but it also includes knowing why those methods work—how they can organize, clarify, and generalize information to a wider world of possibilities. In other words, we must try to give students better and more representative samplings of the comprehensive understanding that an expert has.
This is a noble goal, which undoubtedly the majority of math educators share. But how do we get this to really happen?
The most common method, historically, aimed to do this by immersing the student in a rich bath of procedural know-how splashed with mostly classroom word problems of various types—not teaching specifically or long enough to achieve that deeper holistic understanding, but hoping instead that with sufficient exposure to the procedures (through repetition and taking enough classes), most students would eventually acquire it. But in the mind of some well-respected authorities, aiming to produce genuine understanding this way is a “forlorn hope.”12
This historically common method has always been and continues to be the target of some of the harshest sustained criticism leveled at math education, which has spawned many large-scale reform efforts throughout the last two centuries to improve upon it or do away with it entirely. These efforts, from both inside and outside of the profession, are still ongoing.
Teaching the subject to students this way, however, has not been the unmitigated disaster that it has sometimes been portrayed to be. In fact, many students actually do learn a good amount of algebra and often become quite proficient with the algorithms—certainly more now than a majority of their counterparts two or three centuries ago: some to the point that they actually end up liking and appreciating the subject and going on to take more courses.
Nevertheless, the method has real issues if so many students leave the classroom with a healthy resentment for the subject rather than an appreciation of what it can do. Even for those students who like algebra, success in class does not necessarily translate into real meaning outside of the classroom for many of them. People like and play board games well too, but in most instances such games have no meaning outside of the context in which they are played.13
Moreover, far more disconcerting is the fact that not only does such teaching not get most students to the levels of insight that we want, but it also leaves a larger portion of them with the diametrically opposite view of mathematics than we want them to have—which is viewing the subject as an incomprehensible and meaningless exercise in symbolic manipulation. Something with little relevance to them. Something to be feared and avoided.
All of this suggests that simply exposing people to topics and having them perform exercises is not enough with algebra. In algebra a student can hear an instructor talk about the subject and see them work multiple dozens of examples in front of them, then can participate by doing many more problems themselves (ad nauseam), yet may never experience (on their own) the levels of usable understanding, comfort, and appreciation that we want them to obtain.
THE SUNLIGHT OF WORD PROBLEMS
If we want students to experience the satisfaction of a deeper and more holistic understanding of algebra, the evidence suggests that we have to explicitly teach for that type of understanding up front. Hoping that it will arrive by osmosis for students only through the exposure of having to take required classes (which occasionally hint at it) just does not work for the vast majority of people. In short, if we want students to experience the wonder, satisfaction, and appreciation of algebra, then we should probably put these into our teaching right from the start and make it transparent to them throughout.
However, there are still no guarantees. Even if we are able to find better ways to teach for deep understanding, it is still a tall order to expect that the majority of students will reach the levels of appreciation and comprehension that we hope for in the brevity of time available. Simply put, effective public education for all—even in a perfect world—seems to be a genuinely complicated problem, especially in mathematics where learning abstract and unfamiliar content and norms is hard.
Educators, past and present, are not simply making these difficulties up.
The enormity of this task is further magnified by the fact that some students simply don’t care, have insufficient preparation or bad study habits, often forget what they’ve learned (regardless of age), experience severe math anxiety, test poorly, and are many times also confronting (on a daily basis) the extremely serious issues of life, environment, and home. Adding to this complex soup is the fact that methods that work for one group (e.g., returning adult students) may dramatically founder when used to instruct another group (e.g., ninth grade students).
The pervasive presence of these critical realities should be recognized and appreciated by any who would weigh in on math education—especially those with the ability to influence policy or outcomes.
But the potential for using word problems to aid in comprehension and appreciation is substantial, and excellent instructors have tapped this well throughout the decades. Much of this potential can be reached through the use of classroom word problems. Let’s discuss some of the direct benefits of using these types of problems:
• Such word problems allow us to work with simpler numbers, meaning not only small values but also round or whole numbers, integers, and fractions. This is a distinct advantage over many real-world applications, which can often involve more complicated fractions and decimals and require more initial maneuvers to translate into the language of algebra. More complicated situations can distract students, causing them to focus on other computational and translation issues rather than on the algebraic structure of the problem, which is where we usually want them to be.14
• They generally require less background knowledge than true real-world applications. Problems from science and business often require students to know or be introduced to a fair bit of information about the fundamentals of physics, chemistry, computer science, finance, accounting, or biology before they can understand the essentials in a problem. As students try to synthesize information across multiple fields of study, this can be an obstacle to the primary function of the problem—learning algebra.
• They showcase that it is possible to take a problem in language—often containing inconvenient or difficult-to-obtain information—and drill it down to its quantitative essence, thus allowing us to uncover the key algebraic relationships involved, from which it becomes possible to obtain answers in a systematic, almost recipe-like fashion. Many classroom word problems naturally illustrate this near feat of magic: that of turning mathematical sophistication into routine.
• They allow for the assembly-line production of situations based on many possible variations of a single conceptual theme. This allows students to practice the same algebraic ideas in different contexts as they learn to set up the algebra, simplify expressions, and solve equations. Real-world problems often have a custom-made feel to them that, if exclusively relied on or improperly coordinated, can make it hard for algebraic novices to gain a sufficient enough foothold to acquire the necessary understanding and confidence in the subject for effective appreciation and use. Such production helps us explore how tweaking the algebraic expressions and parameters can give us insight into new situations.
On the whole, because of these properties, classroom word problems can give students a more unified experience in algebra, showing them that the subject is a thought-provoking, highly interconnected, and, dare I say, beautiful entity in its own right.
THE MOONLIGHT OF WORD PROBLEMS
Other, more subtle benefits can also be obtained from the use of classroom word problems. Embedded within such problems are realistic and deep properties of math and nature that can be teased out.
The math subcommittee to the influential Committee on Secondary School Studies (1890s) had something to say on the matter, and though the following was written specifically in their comments on arithmetic, it certainly applies to algebra as well:
The pupil who solves a difficult problem in brokerage may have the pleasant consciousness of having overcome a difficulty, but he cannot feel that he is mentally improved by the efforts he has made. To attain this end he must feel at every step that he has a new command of principles to be applied to future problems. This end can be best gained by comparatively easy problems, involving interesting combinations of ideas.15
I doubt that the educators on this subcommittee, such as Florian Cajori, Simon Newcomb, and William Byerly, were explicitly opposed to teaching students how to use mathematics to solve a customized brokerage problem; rather, they were simply more interested in stressing the potential power for explanation—and the demonstration of method—inherent in the judicious employment of basic problems. In other words, they were advocating that “smart campaigns for deep understanding” (that include a student’s attitudes and impressions) be put in place when using problems in the mathematics classroom.
This thinking around using classroom word problems is similar in spirit to how physicists teach the motion of falling objects in basic classes. Although the drag created by air resistance is a very real effect that must be accounted for to obtain accurate calculations on Earth, they often first ignore this in instruction, as including it can greatly increase the level of mathematical difficulty for most students—to the point of crippling distraction.
The trade-off is that the answer students obtain with these “training wheels” on in effect isn’t as precise as it could be, but that isn’t always the goal of an exercise in the introductory physics classroom. It may sound counterintuitive that the point of solving a problem in this context is not to obtain the most correct solution, and certainly a working physicist would need to prioritize accuracy over simplicity. In education, however, this type of simplification enables students to focus on learning fundamental aspects of motion and forces, and how gravity, accelerations, velocities, heights, and time interrelate, as well as how to systematically use mathematics to deal with such phenomena. The more exact details can be dealt with at a place further along in the student’s education.
This thinking also shares similarities with the way scientists acquire fundamental facts about nature through experimentation. The setups of many of these experiments are often highly choreographed and artificial. And being seen as deliberate staging, such approaches have had their critics—the philosopher Thomas Hobbes among them.16 But because the laws of nature don’t turn off in these cases, even staged and carefully controlled scenarios can be useful tools for illuminating such laws.
Indeed, it is precisely the fact that the laws still work in such cases that allows scientists to isolate a particular property or aspect of a phenomenon from the others in an experiment so as to obtain deep insight. The last 400+ years have shown us the overwhelming potency of employing this investigative method to learn about nature (and this includes Galileo’s and Einstein’s highly productive thought experiments, too).
In much the same way, classroom word problems being used in close coordination can allow us to focus in on various structural things in algebra. It is as if we can make up lots of different little algebraic experiments to shine a powerful spotlight onto some of the deep, fundamental properties of quantitative variation. But these properties must be explicitly pointed out to students, as they generally won’t recognize them on their own; algebra is extremely clever at masking its tracks.
Let’s take a look at a few of the less obvious properties that popped up during our use of classroom word problems:
• Consider the “variable expressions” viewpoint versus the “unknown values” viewpoint: The “variable expressions” viewpoint illustrated that although we initially thought we were solving only one specific word problem, the problem actually turned out to be just one instance of an entire family of rhyming word problems (see Word Problem 3 in Chapter 5). More generally, this viewpoint provides an avenue to a smoother transition to the function concept, which becomes so important elsewhere in mathematics and in the physical sciences.
The “unknown values” viewpoint helps us to see that although we may not know the value of a given number, we can still incorporate it into expressions and manipulate it—as if we did know the number—thus projecting our knowledge, in some cases, far beyond what we knew at the start of the process.
• Drilling down to the algebraic essence of problems showed us that many situations, though dressed up differently, have quantitative similarities. The problems in the last chapter involved unknown measures of angles, ages, different types of coins and bills, and rope lengths. Yet when we distilled them down to their quantitative essence, we saw that they were all solvable by highly similar algebraic expressions and equations.
• Simplifying algebraic expressions and equations shows how variation and stability collectively mix together. We may have several quantities of assorted types in a single problem that are changing together simultaneously, and in our quest to understand what is going on, natural questions arise about the net result of their interaction.
The basic rules of algebra, which allow us to tag different types of variation and combine together those of the same type, can then be employed to significant effect in giving us the ability to better understand and sort out this collective behavior (see “Separating Out Numerical Interactions” in Chapter 2).
• Algebra strengthens arithmetic skills. Practice makes more perfect, and in order to do algebra, one must necessarily do the arithmetic embedded in the process. Many students who enter an algebra course struggling with their numerical skills—especially in regard to integers and fractions—emerge on much more solid computational ground, even if they still haven’t fully grasped the algebra itself. Put another way, we don’t stop learning arithmetic simply because we have started studying algebra.
• We can get a lot of bang from very simple combinations of variations represented only by multiples of x (6x, 55x, etc.) and basic numbers (–5, 10, 17, etc.). Word problems come in infinitely many faces and forms, yet multitudes of them can be described through the use of basic terms involving only numbers and multiples of the simplest variation represented by x (such as 5x + 20 = 180 or 76x = 17252).
Though such descriptions are certainly not exhaustive, there are a wealth of situations involving more complicated variations like x2 or 10x—the range of what can be produced from mixtures of just these two simple components is amazing.
CONCLUSION
Throughout this book, we have explored a wide assortment of problems, situations, concepts, metaphors, and story lines. Yet out of that medley, a “grand play” has slowly emerged and coalesced into something that we can now characterize as a bona fide algebraic experience, including the following:
Variation: Observing a numerical ensemble in a multitude of guises.
◌ A central procedure or theme that produces diverse numbers for different individuals, organizations, times, locations, and so on.
◌ An unknown piece of quantitative information that satisfies certain conditions.
Symbolic Representation: Capturing the essence of the variation or conditions with an algebraic expression or equation.
Maneuvers: Simplifying the expression or solving the equation to gain insight.
Analysis: Interpreting the results of the representations and subsequent maneuvers.
Though this summary captures key points in the process, it doesn’t reveal the entire scope of what most educators would like this basic algebraic experience to also consist of.
What does this really mean? Imagine that you’re planning a vacation to the American Southwest. What are the elements that will make your vacation a true vacation? Some might fixate on details like making reservations for hotels and restaurants, packing the car, and visiting national parks. Others might point to the logistics of how travel happens, like using a GPS, keeping the cooler stocked, and making regular stops for gas. However, though these all are definitely aspects of vacationing, something is missing if we only define a vacation by these terms alone. Doing so would omit the qualitative experience of traveling and your motivation for doing so in the first place, be it rejuvenation, recreation, the desire to experience something new, or witnessing first-hand the aesthetic beauty of the spectacular scenery in canyon country.
So it is with algebra. Though the four components listed earlier are essential to the algebraic experience, they don’t singularly convey the reasons why most mathematics professionals want to keep algebra in the curriculum, which include teaching students
• to see how basic algebraic thinking and expressions can help clarify what’s really going on in confusing or abstract quantitative situations,
• to see how the algebraic way offers insight into numerically varying phenomena in general,
• to gain technical confidence and comfort with algebra as a tool for understanding,
• to make illuminating connections that generate excitement and gratification in understanding something new,
• to genuinely appreciate the scale and beauty of algebra and mathematics, and their connections to the world at large,
• to acquire the necessary skills for further exploration of mathematics in other classes or on their own.
It is quite remarkable that a huge chunk of this can probably be accomplished through just the use of carefully coordinated classroom word problems and situations alone—with the accompaniment of explicit and guided instruction on what is happening. And, if so possible, why not use such problems to do much of the heavy lifting, as the quote attributed to Clausewitz figuratively suggests?17
The internal consistency and interconnectedness of classroom word problems, however, can’t be all there is to it. If so, then this algebraic experience—as neat as it can be in cloistered form—would still probably not be deserving of the central place that many educators want it to hold as a required subject in the K–14 mathematics curriculum.
Still more is expected and demanded of a subject that is mandated to touch armies of students—meaning that at some point this experience must be shown to be somewhat relevant in areas that we might actually encounter outside of the classroom. Think of the internal workings of a modern flat-screen television. The coordination of millions of thin-film transistors combined with light and color technology is truly a wonder. But if all these devices were useful for was the wonderful things happening on the inside, most of us, outside of some scientists and engineers, would give nary a thought to them.
The thing that makes tens of millions of us care about flat-screen televisions (and want to bring them into our homes in multiples) is the fact that the successful operation of their internal workings opens wide to us the vast, complicated, and diverse world out there—offering instantaneous, high-resolution access to images from all over the globe as well as entertaining dramas, news, sports, commentary, documentaries, and so on that, taken together, have had an enormous impact on our lives.
And while algebra may never be as popular as television, many of its concepts and techniques are not confined to the inside of the classroom, either. They live outside of it in powerful, and often subtle, ways that have dramatically changed the collective lives of human beings, for better and for worse. Moreover, the methods exhibited in this “grand play” offer us the chance for better awareness and insight into the steady stream of quantitative information that we face on a daily basis—if we only tune in more to what they can tell us, that is.
In the next few chapters, we’ll tune the algebraic antennae a bit more to better listen in on some of the quantitative variations out there—variations that we may personally encounter or hear about in the course of our daily lives. But first, we explore a bit of the history of the quest to introduce algebra into the public school curriculum and the ensuing efforts to keep it there.