Using Diagrams and Manipulatives to teach the Sine Law's "Ambiguous Case"

Looking over my blog and realizing that I neglected to reflect on our lesson on manipulatives, I thought I would use this post to discuss one particular application about which I have been thinking a great deal. I had my first real day in a school last week and I sat in on two math classes, one grade 9 and the other grade 11. In the grade 11 class (a fairly large one), the topic of the day was the Sine Law and the "ambiguous case". For those who may be a bit rusty, the ambiguous case refers to the situation where more than one triangle can be constructed from the information given (2 sides and an angle). Mathematically, the ambiguity can be seen from the fact that sin(x) is not one-to-one, where x ranges from 0 to 180 degrees. Some of the students in the class had difficulty grappling with the idea of there being more than one solution. Although I was mostly watching, bits of conversation I heard made me think that the deeper issue related to the qualities of sin(x) as a function. Although its not a term they would use, the students seemed to be assuming that sin(x) was bijective. I was trying to think of ways to discuss the Sine Law and the different cases (no solution, one solution, two solutions) as simply as possible. The diagram as I've often seen it drawn shows the fixed side and angle, with the two possible fixed-length sides superimposed.

I included the red line in order to emphasize that the triangle outlined by the green lines is an isosceles triangle. The difficulty with the above diagram is that although there are only two triangles of interest, there are six triangles that appear. This leads to confusion with respect to where we should be looking, and what each line represents. One diagram I prefer, though I can't find many examples of it online, is to build the three cases from the intersection of the two black lines show above with a circle.

Shown above are the same two black lines as before. Three different segments were chosen for the second fixed-length side. Imagine that these were rods rotating on an axis fixed at the upper vertex of the triangle, I traced out the circle that the free end of each rod would draw. The intersection of the horizontal black line with any given circle marks the position the rotating segment would have to make in order to complete the triangle. The blue circle never intercepts and so has no solution. The red circle touches the line tangentially at one point and so has one solution. The green circle has two intercepts and so has two solutions (the ambiguous case). I could have more thorough and drawn a fourth, larger circle that also has only one solution.

I like this diagram because it contains a lot of information yet presents it in what I feel is the simplest and most natural way. The fixed properties remain fixed and it is clear that the only variable is the orientation of the second side. Practically, this diagram could be made in a few ways. I would be drawn from scratch by a student using a compass and a rule. What's nice about this approach is that as the circles get larger and larger, the nature of the solutions changes. This would lead to a natural path of investigation through which the student can be guided into deriving expressions that define each of the four cases. More tangibly, the black lines could be drawn on a piece of construction paper with a marker. The length of the second side could be chosen and a compass could be used to find the orientation(s) of that side. Alternatively, the side could be made out of strip of construction paper and rotated by hand until a solution is found. In whatever approach is used, it would to a good demonstration of the Sine Law and the ambiguous case, as well as connection between calculation and measurement.

“forgottenbooks.com” as a Historical Teaching Resource

This week­ we looked at the idea of using games in order to teach mathematical concepts. It began with a round of “headbands”. Normally when this game is played, each player has a noun taped to their forehead and they play ’20 questions’ to try and figure out what it is. In this version, instead of a noun, we had a quadratic function. This was an interesting spin on the game. I don’t know how other people played, but for myself, I first figured out the form of the function (factored, vertex, expanded) and then guessed the parameters. I’m not sure how much could be learned from this game but it is certainly a good way to test your facility in mathematical vocabulary. For example, if I ask “is the function in vertex form”, does the student know that is actually saying “does the function look like f(x) = a(x-b)²+c”? We went from there to discuss different video games and apps that have been developed for the teaching of mathematics. I reserve a slight amount of skepticism with respect to how much can be absorbed passively from playing these games, but as a matter of practicing a concept, I imagine a student will get through many more practice problems if they are disguised as games (chocolate-covered broccoli, as Amy Lin put it). The side of technology that really intrigues me is its use as a virtual manipulative. For example, apps that allow for an intuitive visualization and manipulation of geometric figures would have a useful degree of flexibility that their tangible counterparts would not have.

Since we have a whole forum dedicated in this class to the discussion of different apps, I want to bring up a potentially useful but quite different type of resource.  There is an online PDF library called “Forgotten Books” which contains literally hundreds of thousands of rare historical books on every subject imaginable, from Renaissance literature to essays on witchcraft. Membership can be free or purchased (about $8 a month). The free membership gives unlimited access to all of the books, but every couple of pages the image will be blocked. The paid membership removes the barrier. I bring up this site because it contains many historical works on mathematics as well as math pedagogy. These books present a very interesting perspective and explain concepts in a way that may be unexpected and possibly useful to a teacher. They also make reference to different manipulatives that could be of interest to the modern reader. These are often simple, and could be made by hand without much difficulty. They would also make good projects for someone who would like to try 3D printing. These books can also serve as sources of interesting and instructive word problems as well as ideas for simple yet effective demonstrations of mathematical concepts.

Another use for this library is as a source for historical perspectives on mathematics and pedagogy. These books being so old (+100 years), I find this perspective very refreshing. At the same time, it is very interesting to read a textbook from more than a century ago and see that many of the concepts we credit to modern enlightenment such as “student-centered learning” (and even differentiated instruction, to a degree), were practices treated as merely common sense. In fact, the more I read these books, the more I feel that the derision “traditional” education faces these days is built on a straw man, but I digress. One part of the library I would particularly like to explore are the books on Jesuit education and their pedagogical philosophy. This is not something I know much about, but the historical importance of the Jesuits on the scientific revolution was first impressed upon me when I read Rene Descartes’ biography, Cogito Ergo Sum. In addition to training many of history’s most important mathematicians and physicists (including Descartes), they also educated Marin Mersenne, a French priest who single-handedly organized and maintained correspondence and collaboration between nearly 140 different mathematicians, philosophers, and scientists from all across Europe (a thankless task whose historical importance could not be understated). In conclusion, Forgotten Books is a very fun site to explore. The sheer variety of books available is such that you never know what you will find when you browse, but you will always find something worth reading.

Gradus ad Parnassum

In my previous post, I wrote briefly about a historically important pedagogical work in music theory entitled Gradus ad Parnassum, or in English, The Steps of Parnassus, (Parnassus being the mythical home of the Muses). This work first appeared in 1725 and has never fallen out of use, being used by composers and teachers ranging from Mozart to Brahms to Richard Strauss. The author was Joseph Fux, a major composer of the late Baroque period. Although his fame as a composer has not survived as well as some of his contemporaries, J.S Bach and Handel being obvious examples, he was highly admired in his own time and occupied during his career the most prestigious musical post in the world: Music Director of the Viennese Court. I would like to speak in more detail about the Gradus because I feel there is much that we can learn from Fux’s approach that can be applied to the teaching of mathematics and physics. It may be surprising to some that a parallel could be drawn between mathematics and music pedagogy, but the actual art of composition is more systematic that some would think. It is part art and part science, much like mathematics itself.

In any math problem of a particular level of sophistication, it is the general aim of the student to arrive at the correct answer.  In other words, there is some unknown value or function (in the case of a differential equation) and the object of the game is to derive an expression for the unknown. This is most systematic level of mathematics. For example, given a Hookean spring with particular characteristics attached to a mass with certain initial conditions, a differential equation can be written from which the motion of the spring can be obtained. Beyond obtaining that function, there is little that can be drawn from that problem. One step above in sophistication would be to study the problem more generally, studying the system while directly specifying as little as possible. This would enter the region of mathematical proof (as in, to prove that the period of a Hookean spring does not depend on the amplitude of the oscillation). It has been my experience that this kind of mathematics has something in it akin to art: we take certain facts and try to arrange them, or follow their implications in such a way that a whole new fundamental truth is revealed, not merely the specific truth or a particular system with particular values. In music it is somewhat similar, to arrange sounds and pitches in a new and meaningful way. The difference is that in mathematics, the aim is consistency, which is objective; in music, the aim is aesthetic appeal, which is partially subjective. Music theory is the system of rules and conventions that attempt to formalize ‘correctness’ in composition. Unlike in mathematics, these rules are not unbreakable but rather “for the guidance of the wise and the obedience of fools”. By this I mean that good composers break these rules when they desire a particular effect, while bad composers break them because they lack the skill to follow them.  Turning now to the Gradus ad Parnassum, Fux outlines his pedagogical philosophy in his introduction:

There have certainly been many authors famous for their teaching and competence, who have left and abundance of works on the theory of music; but on the practice of writing music they have said very little, and this little is not easily understood. Generally, they have been content to give a few examples, and never have they felt the need of inventing a simple method by which the novice can progress gradually, ascending step by step to attain mastery in this art. I shall not be deterred by the most ardent haters of school, nor by the corruptness of the times … Seeking a solution to this problem, I began, therefore, many years ago, to work out a method similar to that by which children learn first letters, then syllables, then combinations of syllables, and finally how to read and write. And it has not been in vain. When I used this method in teaching I observed that the pupils made amazing progress within a short time. So I thought I might render a service to the art if I published it for the benefit of young students, and shared with the musical world the experience of nearly thirty years, during which I served three emperors (in which I may in all modesty take pride).

The work itself is written as an ongoing dialogue between a pupil Josephus and the teacher Aloysius (representing the Renaissance master Giovanni Pierluigi da Palestrina). Aloysius begins with a review of consonances and dissonances (which he assumes the student to already know) and then proceeds to introduce the three kinds of motion: direct, contrary, and oblique. He concludes the introduction with the four fundamental rules of counterpoint:

1.     From one perfect consonance to another perfect consonance, one must proceed in contrary or oblique motion.

2.     From a perfect consonance to an imperfect consonance, one may proceed in any of the three motions.

3.     From an imperfect consonance to a perfect consonance, one must proceed in contrary or oblique motion.

4.     From one imperfect consonance to another imperfect consonance one may proceed in any of the three motions.

These rules form basically all the information that the reader (living vicariously through Josephus) needs for the first lesson. Aloysius presents the student with a melody (or cantus firmus) and has him harmonize it in simple note-on-note counterpoint. This is the musical equivalent to a simple word problem. The fictional student presents his answer and explains to the teacher how he arrived at it. However, the student’s answer is not perfect and Josephus corrects it. Some of the mistakes are from breaking one of the Four Rules. Some mistakes are subtler, such as not setting the counterpoint in the same mode as the cantus firmus.

When I read this example, I thought at the time that it seemed somewhat sloppy, throwing information at the student after the fact. However, speaking for myself, the phrase “set the counterpoint in the same mode as the cantus firmus” was easier for to understand from the mistaken example than from the rule itself. The student’s work then becomes an instructional example, with the teacher demonstrating what he could do to avoid the error. This is the general pattern of the book. Very small amounts of information are introduced at a time, which allows for a steady but never overwhelming increase in complexity. The work progresses while remaining in two-part harmony, through the first, second, and third species of counterpoint, ligatures, and florid counterpoint. Having exhausted this type of composition, Fux then goes back to the beginning and does it all over again in three-part and four-part harmony.

When I read this book, barely more than 100 pages, I wonder what specifically it is that made it so effective. What struck me first was how old fashioned the compositional style being taught was. The Baroque period was coming to a close at the time of writing. Renaissance polyphony, which had been essentially replaced with Baroque counterpoint, was quickly giving way to the Rococo style. What Fux was intending with this book was to solidify the old way of doing things, not because he felt that people ought to write that way, but because he saw that the young composers were ignorant of the historical foundations of their art and were becoming reckless and unbounded by any standards of taste. He says this explicitly in his introduction where he states that that his purposes “do not tend – nor do I credit myself with the strength – to stem the course of a torrent rushing precipitously beyond its bounds. I do not believe that I can call back composers from the unrestrained insanity of their writing to normal standards.” Music pedagogy then, was either too theoretical or heavy with information, or it was too undisciplined, prompting a great deal of activity yet resulting in an overall decline of quality and technical mastery.

I wonder sometimes if these two errors are still present to some extent in mathematics (and physics) pedagogy. I’m always struck when I tutor high school students what tomes their textbooks are, and how difficult it is to find what I’m looking for in them. This would be an example of too much information, particularly when the underlying concept is very simple. It frustrates me enormously whenever I work with a first-year university student, and find that after two years of high school physics they still can’t do any kind of projectile motion problem without flipping back and forth through the textbook, or define for me Newton’s 2^nd Law or the Work-Energy Theorem (and this happens a lot). The textbook almost functions for them as a book of spells instead of an instructional tool. The other extreme is when creativity and discovery is emphasized at the expense of formal instruction, in the expectation that students through exploration will spontaneously discover the principles they need. (Re, “We Don’t Want No Education” by Theodore Dalrymple). I expect that in some circumstances this may work very well, but I wonder at the same time if it can allow serious misconceptions slip past the teacher. Also, I wonder if making a subject appear more subjective that it really is can inadvertently encourage in students a relativist epistemology. Whatever the virtues of the present age, I don’t believe due respect to Principle of Non-Contradiction is one of them.

I believe that part of the brilliance of Fux’s text is the skill with which he takes the middle road. In modern parlance, it is very ‘student-centered’ in that the student in the dialogue does the bulk of the work, and information is introduced very gradually. There is no deluge of information but rather, knowledge of music theory is grown naturally in the student through the use of exercises. At the same time, Fux does not beat about the bush with respect to the student’s ignorance of the subject. He presents his rules and methods as law and not as a matter of debate. He sometimes explains his rationale to the student and sometimes refuses to, if he feels the explanation is premature or would confuse the student more. At one humorous point in the dialogue, Aloysius advises Josephus not to get over-confident and to remember his place:

I want to remind you again and again to make every effort to overcome the great difficulties of the study you have undertaken; and neither to become discouraged by hard work; not to allow yourself to be deterred from unflagging industry by flattery of such skill as you have already achieved. If you will work thus you will be delighted to see the way in which light gradually illuminates had been obscure and how in some manner the curtain of darkness seems to be drawn away.

To conclude, part of me wonders what a math or physics instruction book would look like written in this manner, and if it would even be possible. Being a dialogue, it is not particularly useful as a reference, but I think this format makes it especially effective as an instruction tool. It is possible to read a textbook and come away thinking that you’ve learned something: perhaps you have, and perhaps you haven’t. However, it would be just about impossible to read Gradus ad Parnassum and convince yourself that you’ve learned something if you haven’t also done the exercises. Yet history shows us that someone who diligently works his way through the entire book could gain an enormous amount of skill and understanding. As such, Fux’s work is not so much a textbook in the modern sense but an actual pedagogical program. I won’t speculate on how effectively this method could be transplanted from music theory to mathematics, but as someone who has had to learn both, the natural similarities between these two subjects make me believe that anyone who wants to learn how to present mathematics in an intuitive guise and a sensible order may glean many useful ideas from this work.

The Test of Excellence

This week we were required to read an interesting article on math pedagogy entitled “Relational Understanding and Instrumental Understanding” by Richard Skemp. In the article, Skemp makes his case that too much math pedagogy is based on teaching calculation techniques and procedures (what he calls “Instrumental” understanding) as opposed to instilling a deeper understanding of mathematics, from which it should be apparent to the student why those techniques (“Relational” understanding). His thesis, if I may so condense it, is that students are rarely being taught actual mathematics but instead of simply memorizing computational rules and the situations in which they apply. Students then can correctly do specific tasks very well, but if presented with something they haven’t seen before, will be at a loss on how to proceed.  In principle, I very much agree with his concerns and have seen the equivalent problem in the first-year physics students I used to TA, who often viewed physics as little more than applying a succession of mysterious formulas of dubious origin. However, I am less enthusiastic when I imagine what would be required in order to implement Skemp’s ideas effectively, and what the outcome would be if the attempt failed.

There is something dangerously optimistic about viewing prospective public policy through the lenses of the best-case scenario. I am sure Skemp himself would be very effective at teaching his own methods, not least because he is an actual mathematician. However, a teacher cannot teach what he does not know, and I wonder what would happen if a teacher who had little to no background mathematics was made to teach in this manner, as though he were an expert. Speaking for myself, I believe I could teach math and physics in this way (and I intend to), but I am not sure that I could do it with chemistry. This is not because I don’t know anything about chemistry, but because I don’t have nearly the same level of expertise in it that I have in physics. I have observed this pedagogical approach go awry when I tutored high school students in physics. I had one student who was struggling quite a bit with relative motion. The student showed me diagrams the teacher was having them draw in order to describe different kinds of reference frames. It was a very “relational understanding” approach, and perhaps could have been effective. However, from the student’s course notes, copied from the board, I could see that proper use of the diagram had been explained incorrectly and that by making these diagrams, the student had been reinforcing a serious misunderstanding of the material. This would take a lot of work to fix because the error ran so deep in the student’s understanding. Thinking of experiences like this, the question that comes to my mind when I read this kind of pedagogical theory is: used well, this method promises certain obvious benefits, but what is the damage if used badly?

I am naturally resistant to a wide and enforced implementation of ideas, even ideas I agree with. This is because ideas have unintended consequences, and if some of the consequences are bad, a broad implementation will make them widespread instead of local and more easily addressed. Additionally, it is often the weakest and poorest members of society who are least equipped to cope with the consequences of bad policy. I recently read a very sobering article by the British psychiatrist Theodore Dalrymple, who worked for most of his career as a doctor in an English prison and in a neighboring slum hospital. He has produced a huge body of very depressing essays and articles describing the disastrous social consequences of ideas that begin at the top, trickle down, take root, and then run amok. In his article “We Don’t Want No Education”, he described some of his countless observations on the current educated state of his teenaged patients, part of which he blames on experimental educational practices:

… I now test the basic literacy of nearly every such youth I meet, in case illiteracy should prove to be one of the causes of his misery. (I had a patient recently whose brother committed suicide rather than face the humiliation of public exposure in the social security office of his inability to read the forms he was required to fill in.) One can tell merely by the way these youths handle a pen or a book that they are unfamiliar with these instruments … I cannot recall meeting a sixteen-year old from the public housing estates near my hospital who could multiply nine by seven (I do not exaggerate). Even three by seven often defeats them. One boy of seventeen told me, “We didn’t get that far.” This after twelve years of compulsory education (or should I say, attendance at school).

 http://www.city-journal.org/html/we-don’t-want-no-education-12348.html

I don’t think situations like this could flow immediately, or even naturally from a single idea or theory, but I do think that a good idea for a particular situation can become a very bad idea when pulled and stretched into a broad catch-all policy. Skemp’s theory, and theories like it I feel are abused when they used to downplay the importance of calculation. The question is often raised, why bother focusing on hand written calculation when we have calculators? Why not just focus on “understanding”. What strikes me most about this question (as well as the related, “why bother with spelling when we have spell-check”), is the way in which these skills are so carelessly disposed of, as though the mental development that comes from mastering these basic skills isn’t reason enough to teach them, as well as the enrichment of our lives that can only come from filling our minds with meaningful things. Why learn facts about history if you can just look it up when you need it? Why memorize any great sayings from poetry or philosophy when we have Wikiquote? This philosophy, though easy and convenient, seems to me to lead to a cold and stifling materialism. It is in this respect that I fear that we as a society are becoming a bit like Lady Bracknell, who disapproved of anything that interferes with natural ignorance. Setting aside the question of the intrinsic value of knowledge, at the heart of the pro-calculator stance there sits an assumption I’d like to address with respect to the purpose of calculation.

I occasionally observe an attitude towards the importance of calculation that I think misses the point, namely, that it is an end in and of itself. In any level of physics or mathematics, do we give students problem sets so that they can practice calculations, or rather so that by doing the calculations, they will further cement the foundations they are being taught? To give a simple example, a common beginner physics problem is to describe the motion of a box sliding down a ramp. Typically some variables will be given and the student has to use Newton’s Laws to calculate the speed of the box, or the coefficient of friction, or the angle the ramp makes with the ground etc. A student in high school or in the first year of university will see many problems like this, all alike but slightly different. At the same time it can be shown algebraically that these problems are mathematically identical and that, having solved it once, you’ve actually solved every permutation of it as well. What then is the point of doing the same thing over and over again? It is to force the student to re-derive the algebraic result, but each time within slightly different context, or from a different direction, hopefully deepening their understanding in the process. The students can be shown (and I do try to show them) that they are in fact doing the same thing over and over again, but it takes a lot of thinking and a lot of practice in order to see it. Once you reach that point, I would say that you truly understand it. In this case, the arithmetic involved is secondary, and it would not defeat the purpose of the exercise to use a calculator. (As an aside, I’ve directly observed many students whose ability to do these problems is impeded by their reliance on calculators, because instead of working through the problem algebraically and then putting in the numbers, they just number crunch the entire time. If they get the wrong answer, it’s impossible to see where they went wrong because their work up to that point is just a meaningless list of numbers.)

This kind of physics problem builds on top of skills that should already have been learned. In particular, arithmetic, solving sets of linear equations, and rearranging equations. To learn each of these skills, you would have to go through the same process described above, but at a lower level. You would learn the theory (hopefully taught meaningfully, as Skemp would advocate), and then practice that skill over and over, from different angles, cementing your learning and fixing your misconceptions. For learning arithmetic, using a calculator would completely defeat the purpose. If the point was to simply do the calculation, it could make sense to use a calculator since it may be faster and less error-prone (though for really simple problems, I’m not so sure). But we do not learn arithmetic to do simple calculations, we do calculations to assess and develop our arithmetic, and to ensure that the foundation is in place in order that we may proceed to more advanced topics.

I’ve seen this kind of thinking before, albeit in a very different context. One of the most important pedagogical works in the history of music is Johann Fux’s Gradus ad Parnassum from 1725. This book was used to instruct many of histories greatest composers including Mozart, Haydn, and Beethoven. J.S Bach, perhaps the greatest musician of all time, strongly approved of it. I would like to write about it in more depth in a later post but for now, I will note that the structure is written as an actual dialogue between a teacher and a student. The teacher explains the principle and the student does a series of short exercises. After each exercise, the student explains what he did and why, and the teacher approves or corrects them while explaining the reason for his mistakes. Each chapter builds on the one that preceded it, until by the end, you are writing in full four-part harmony. It’s a remarkable approach, but what strikes me about it is the balance Fux achieves in combining theory and conceptual understanding with practice. Sometimes I think we feel that if the understanding is strong enough, correct usage will follow, but in my own experience as a student, this was rarely the case. They feed and correct each other. Fux states this explicitly in his introduction:

You will notice dear reader, that I have given very little space in this book to theory and much more to practice, since action being the test of excellence, this was the greater need.

Similarly, to take a more contemporary example, J. J. Sakurai in the foreword to his graduate level textbook on quantum mechanics states, “the reader who has read the book but cannot do the problems has learned nothing”.  I think most people would agree with this sentiment when applied to advanced science, but why is it a point of contention when applied to elementary math? I would maintain that the student who has been taught arithmetic but needs a calculator has learned nothing since, as Fux said, action is the test of excellence.