# Restoring the Practice of Subdivision

This scenario is not unlike how students often perform rhythms in an ensemble. Each child has only a conductor’s beat “marked,” and each student guesses at how divisions of the beat should be played. If a student has learned rhythms solely with rhythm syllables without learning how those syllables relate to a pulse, then they can only guess at where to place those beat divisions as they perform. Every students will do it a little different, and the result when they all sing or play together is that the performance is rhythmically messy and inaccurate.

Every rhythm is a product of durations performed to a pulse. Gordon has called that pulse a macro beat, and others have called it an ictus. Conductors and students alike work very hard to communicate and follow, respectively, that ictus, given by the conductor’s time-beating motion. But “watching the stick” is simply not enough. Those following the conductor must be able to accurately perform what comes between those conductor beats. Noted band conductor Frank Battisti once said that a conductor’s responsibility is what happens on the beat, a players responsibility is what happens between the beat. How do we teach our students to handle that responsibility? The answer is that we must teach them to be keepers of two kinds of beats simultaneously: the macro beat and the micro beat, the latter of which is the first division of the macro beat. In common time, there are usually two micro beats (two eighth notes) for every macro beat (a quarter note). In so-called compound meters such as six-eight, there are usually three micro beats (three eighth notes) for every macro beat (a dotted quarter note). As students “watch the stick,” they must also be audiating even eighth notes (micro beats) in order to play what comes between the beats accurately.

It has been my observation that conductors do not bring this up until a difficult rhythm is encountered. Only then will they tell their students that they must “subdivide” in order to play accurately. While this is a sound remediation, the fact is that subdivision should be going on all the time. It is part of fully understanding and perceiving any piece of music. It helps groups of musicians play quarter notes together as surely as it helps them play intricate divisions of the beat. What’s more, it doesn’t require much teaching or practice to be able to subdivide. All a conductor needs to do when he or she hears the rhythmic stability start to falter is to begin conducting or tapping out the subdivision, and usually the accuracy will snap into much greater precision. The ability to subdivide, or to audiate micro beats is innate. It is drawn from how the various durations in music are naturally organized by our brains into patterns that are subdivided. Our job as music educators is to give our students as much experience with a variety of rhythms as possible.

Earlier, I mentioned the importance of associating rhythm syllables to a pulse. Rhythm syllables that are merely recited phonetically without regard to an ongoing pulse will not bring about effective rhythmic learning. Simply calling a pair of eighth notes ti-ti in the absence of an audiated ta will not transfer to music literacy. Similarly, trying to explain rhythm by telling students something to the effect that eighth notes or ti-ti’s go twice as fast as quarter notes or ta’s leaves the questions of how fast is twice as fast, and twice as fast as what, unanswered. With the syllables ti-ti, the first ti is the macro beat, and both together are the micro beat. One must hear the first ti in each pair as the ictus, and the second as what is going on between the beats, placed exactly even between the preceding ti and the following one. This is why I prefer syllables that differentiate between notes that are macro beats and those that are not. It helps the student maintain an understanding of what he or she is doing throughout. In Gordon’s system, for example, instead of ti-ti, there is du-de. Du is always the macro beat, and du-de is always the micro beat. Students know that no matter if there is an eighth note following the ictus or not, that ictus is always du and the eighth note that follows is always de; two different sounds for two different rhythmic functions.  (For a further explanation of rhythm syllable systems, see my articles on the subject elsewhere in this blog.)

With “what happens between the beats” firmly in the mind’s eye, students will quickly become more accurate in their rhythm performance. Subdivision should be a constant and ongoing operation for all musicians, not just an occasional remedial strategy. Subdividing while listening to music also enhances enjoyment and understanding, because the rhythmic structure of the music to which students are listening is revealed to them through the accurate realization of duration to beat relationships, and resulting patterns of strong and weak beats which constitute meter. Subdividing fosters greater musical success and enjoyment.

# What Is Musical Dissonance?

When I was a high school student, I was sure I knew what dissonance in music was. If it sounded wrong, it was dissonant, and if it sounded right, it was consonant. An interval of a 2nd, or a try tone, or a seventh was dissonant, and all the others were consonant. Then in college, I learned that a perfect 4th is dissonant, though it still sounded fine to me. But that was the first hint that something was amiss with my definition. If an interval that sounded right to me was dissonant, then I needed a new understanding of the concept. In my discussion of expectations (see “Is All Music Intended to Be Expressive?”), I mentioned Meyer’s thoughts on continuance and repose. Music that demands continuance, or “leaves us hanging” creates tension, whereas music that comes to rest harmonically, as at a full cadence, expresses repose or relaxation.

Once of the expectations Western listeners have is that tension will resolve into relaxation. We are accustomed to patterns of tension and relaxation in our music. This is the basis for traditional voice leading and harmonic progressions. Ornaments such as suspensions, appagiaturas, trills, and so forth create tension because they sound incomplete, as one who stops in the middle of a sentence. We know from our experience with grammar that there has to be more to that sentence, and we know from our experience with music that there has to be more to that musical phrase, more music that will bring us to that cadence, that musical punctuation of a comma, semi-colon, or period. It is that expectation, even necessity of the music continuing on to a more suitable pausing or stopping place that makes the present moment sound dissonant.

In framing dissonance in this way, I am essentially equating dissonance with continuance. The unresolved suspension is not dissonant because it sounds “wrong,” it sounds dissonant because it leaves us demanding more. Dissonance is unfulfilled expectation. It is a form of anxiety similar to what we experience when we worriedly await the outcome of some life event. That anxious, stressed feeling is akin to how our body reacts to dissonance in music.

Zatorre and Blood (1999) at McGill University created original melodies containing dissonant and consonant patterns of notes, and played them for ten volunteers who were  scanned at the same time. Rejecting the null hypothesis, dissonance made areas of the limbic system linked to unpleasant emotions light up in the PET scans, whereas the consonant melodies stimulated limbic structures associated with pleasure. In other words, music elicits the same emotional response in the human brain as non-musical events with the same emotional makeup. So our experience of dissonance is larger than a response limited to musical stimuli. Our perception of musical dissonance is a parallel response to other life experiences.

In The Harvard Dictionary of Music we find that dissonance represents the “element of disturbance and tension.” While tension can, as we have seen, be framed within unsatisfied or delayed expectations, disturbance is a dangerously subjective idea. Listeners are disturbed by different sounds to different degrees. We must also consider that what was once considered dissonant to musicians and audiences alike are now accepted as less so or even consonant now. There is a phrase in Walton’s Belshazzar’s Feast in which the choir sings the text “drank from the sacred vessels.” “Drank from the” is set to a minor 2nd, “sacred ves-” to a major 2nd, and “-sels” to a minor third. Though the 2nds are by some definitions dissonance, they are to my ear delectable and beautiful in the sense that the imminent resolution is so quickly realized and from the extreme “disturbance” of the minor 2nd. In this, we might consider that dissonance may be determined in part by its duration prior to its resolution. The longer the disturbance remains, the more likely it is to be perceived as dissonance. Dissononances that occupy short time spans may be less apt to be perceived as dissonant because they are more closely associated or attached to their resolutions. This is why a suspension can seem more consonant than an escape tone, wherein the resolution by skip obscures the tranquility of the resolution.

Some have attempted to define dissonance as any interval not included in the prevailing diatonic scale.  As long as diatonicism is the standard for measuring consonance or dissonance, this definition is at least serviceable. But it is rendered inappropriate for atonal works. Hindemith (1900) breached this issue by putting forth a ranking of melodic intervals from most consonant to most dissonant. This ranking was P5, P4, M6, M3, m3, m6, M2, m7, m2, M7, TT. Hindemith believed that consonance and dissonance could be perceived as a kind of floating standard, constantly defined by the current interval regardless of overall tonal center or lack there of. Still, there are vestiges of traditional harmony in his ranking, because the first 4 intervals are all diatonic and all part of the tonic, dominant, or subdominant chords. Hindemith believed that we shifted our perception of tonal center according to intervallic relationships when interval roots were non-diatonic.

This theory allowed for writing in the 12-tone style without abandoning tonality. Tones that are lowest, highest, and longest are given greatest importance in a melodic progression. These tones then are constructed to form step-wise motion, no matter their separation from one another by intervening tones. The interval of the fifth, being the most consonant, is also the strongest harmonically. It’s occurrence over changing roots can thus alter the perceived tonal center, whereas intervals gradually loose their ability to establish tonal centers according to their increased property of dissonance. To state it in terms of our overall discussion, the perfect 5th has the least power of continuance and the highest degree of repose, and so functions as a tonic in traditional harmony. As intervals become more dissonant, they gain greater power of continuance and lessening degrees of repose, and so add tension as well as distance from a perceived tonal center. Listen to this example from Hindemith, and see how much of our discussion you can take away.

Blood, A. J., Zatorre, R. J., Bermudez, P., & Evans, A. C. (January 01, 1999). Emotional responses to pleasant and unpleasant music correlate with activity in paralimbic brain regions. Nature Neuroscience, 2, 4, 382-7.

Hindemith, P. (1900). The craft of musical composition. Mainz: Schott.

# The Other Expectations

Today I would like to discuss expectations, but not the usual sort. Often, when expectations in education are discussed, they are the kind teachers have of students. These may be behavior or performance expectations, and both are important. There is, though, another sort of expectation that is embedded in the how successfully people perceive and understand. These are the expectations a learner brings to that to which they are confronted. Absent expectations, materials presented to students such as whole or part of musical works, can only be understood in a limited way at best. When expectations are incorrect, the musical work is likely to be misunderstood or downright confusing.

Let me use a non-musical example to explain. Suppose a child is looking for a shaker in a box of non-pitched musical instruments, and suppose that child expects to find a shaker that is square. He or she goes through the contents of the box containing shakers, and though many are there, the child overlooks all of them, because none of them matches his or her expectation that a shaker is square. The child finally gives up, and claims that there are no shakers in the box. The child would have easily found many shakers had he or she known that they were round, or egg shaped. When presented with the information that all of the items in the box are shakers, the child will be surprised, and declare “that was unexpected.” And that is exactly the point. If a person’s expectations about what they are presented with are faulty, they will miss the meaning, or even the identity of what they are seeing or hearing.

This point was made by a Music History professor when I was an undergraduate in his class. He played an excerpt from a Mozart symphony ( forty years later, I don’t recall which one), and then asked us what we heard. The usually reliable Bruce responded that he heard violins playing this, and cellos playing that, and so forth. Bruce’s answer was right, but incomplete. I responded that I heard the melody played on the flute over all that Bruce had described. The activity in the strings was what Bruce expected. He was used to the melodic content being delivered by the strings, but did not expect to hear the melody in the flute. His expectation eliminated the possibility of the melody being anywhere else except in the strings, and caused him to overlook it. I have had similar experiences with my students, even adult students. I play them a melody I want them to keep track of in a sonata-allegro form, and then play the movement. What I didn’t tell them was what instrument or instruments would be playing the material I wanted them to hear. Frequently, not knowing where in the orchestra to expect the melody to appear, they listen in the wrong place and miss it entirely. I can always go back and tell them what instrument will be playing it, and (as long as they know what that instrument sounds like) they will easily hear the melody next time through. It is all about knowing what to expect.

The same is true for performers. For myself, when I am playing my clarinet, I can easily play, even sight read, most music put before me, because I expect even difficult passages to be in familiar patterns of scales and arpeggios. As long as those patterns are what I expect, I can play accurately. But the instant the pattern changes, or the scale or chord is one I did not expect, mistakes become numerous, and I am then in a passage I must stop and practice until I have learned those unexpected patterns. As we teach students, it is not enough to teach them the repertoire, or even the scales, which are out of context. We must teach them how these things are typically used in actual music. This can include learning progressions of arpeggios, sequential patterns like scales by thirds, or sequences of motifs. Many etudes typically take this approach, moving through a few themes that use sequence and progressions. They typically start relatively easy, then become difficult somewhere past the middle, and then end relatively easy again. This in itself is an expectation I have for etudes, and causes me to search out the middle of the etude to practice first, expecting that the beginning and end will come much easier. Having these expectations not only helps me play the etude more successfully, but also helps me plan my practice strategy.

Expectations are acquired through experience. As music educators, we provide our students with experience ample for constructing expectations on. I believe that one of the weaknesses of survey type courses, is that there is too great a variety of musical genres, styles and forms presented in an attempt to build familiarity with them all, while not providing sufficient concentration on any one genre, style or form to allow for the forming of expectations of them. No one can have accurate expectations of 18th century symphonic music after listening to single symphonies by Haydn, Mozart and Beethoven. The creative span of just these three composers is so much greater than what can be drawn from a single work. It is not so unreasonable to think that Beethoven’s first two symphonies were written by Haydn, if all one knows of Beethoven is the fifth and third symphonies. Likewise, it is not unreasonable to think that the prelude to Haydn’s The Creation was written by Wagner, if all one knows of Haydn is that prelude. Given the opportunity to listen to a broader sampling of each composer’s work, the listener can acquire more accurate expectations that will guide him or her in perceiving and understanding so much more along any symphonic journey they may take.

Expectations also leaves the listener or performer the freedom to discover and explore musical works with those expectations and the accompanying intuition as guides, so that the performer preparing a musical work for performance or a listener taking in a concert or recording is not left to drift through, become bored and abandon future encounters with such music. Giving students expectations whets the appetite to go out and have those expectations met, and that involves seeking out the musical works about which they have expectations. Developing expectations equips the student to interact with musical works on his or her own terms, without being restricted by assigned listening tasks. Students who make predictions about what they will, concerning everything from instrumentation (what instruments are likely to have the melody often (violins, oboes, flutes, clarinets) and which ones are unlikely to have the melody often (violas)), and harmonic progressions (clear tonic and dominant harmony in Mozart, more adventurous and chromatic treatments in Richard Strauss and Wagner) to length of works and use of rhythm and dissonance. With the right expectations, a beautiful Wagnerian dissonance sounds like a bad mistake in Mozart, and sets that dissonance in the right context, that of impressive creativity on Mozart’s part to think of using dissonance like that (consider the “dissonant” quartet, no. 19 in C major, K. 465)  when others were not doing so. And so the more able we prepare our students to have accurate expectations concerning musical works, the more powerfully they will be able to assert their musical learning on creative musical activity.

I am by nature a very thoughtful person. People who know me well frequently accuse me of overthinking many things, and I have to admit that they are right–I do overthink often. As someone almost constantly in conscious thought about something, there are many thought that come and go, forgotten as quickly as they arrived, but others get my attention. Why are some noticed and others not? Because the ones that get noticed connect to something else I have been doing or thinking, and so are of particular interest to me at the time. The thoughts that remain are those that pertain to what I am at that moment most interested in, what I am presently doing or wanting to do.

There is an important lesson in all of this to teaching and learning, and it is this: in order for learning to succeed, it must be helpful in acquiring something the learner wants. We educators often concern ourselves with goals and objectives, both for our students and ourselves; and well we should. But goals and objectives that are only imposed on a learner, and around which a learner cannot contextualize with relevance are likely to meet with resistance, and be at best of limited use in bringing about the learning we desire. That is why connecting objectives for creating, performing and responding to music is so important, and why it should be done early in any teaching sequence, before the students become mired in trying to achieve objectives that have little or no meaning to them personally. Let’s look at how connecting works in a standards based music classroom.

First, we begin with an enduring understanding. “Musicians connect their personal interests, experiences, ideas, and knowledge to creating, performing, and responding.” When we present concepts, ideas, knowledge, and repertoire to our students, one of the first things they ask themselves if it all seems quite new and unfamiliar to them is, “what does this have to do with what I’m interested in?” “What in my experience with music does this sound like? What associations to familiar things does this music or idea or concept bring to mind? What is something I am familiar with that compares to this?” If the students comes up empty on each or even most of these questions, he or she is unlikely to have any desire to proceed with your lesson or instructional unit. You are about to abandon them to an intellectual deserted island, and that’s not a place where anyone (probably you included) want to be. So before “teaching to the objective” can begin, context needs to be established. Familiar signposts need to be pointed out, and the teacher must give a method of exploring something new in the context of something familiar.

Doing so will motivate learners and deepen understandings. Remember, understanding is not obtaining the ability to recall knowledge or repeat a task, it is the ability to apply previous learning to new situations. Application is only possible when connections are clear. A second enduring understanding states this clearly. “Understanding connections to varied contexts and daily life enhances musicians’ creating, performing, and responding.” Notice what we are trying to help our students connect; we want them to connect varied contexts, the variable, with daily life, the constant. Each new context must connect back to the same personal life of each individual student. Not all students will make the same connections, and not all connections will be equally strong for all students. In fact, one student’s strongest connection, may only be a hint at how to form a different connection to another student, but in an environment of shared learning, students’ connection to varied contexts become woven together, as one connection bolsters up or clarifies another.

I recently played the main theme from the film Indiana Jones to which 2nd grade students tapped a steady beat.  When the music stopped, one child pointed out that the music sounded like Star Wars. He had never seen Indiana Jones, but he recognized something in that music that sounded like music in a movie he had seen, namely Star Wars.  This was, of course, a brilliant connection, because both film scores were written by the same composer, John Williams. With the confidence of having made that connection, that child was now eager to find out what made the two themes sound similar, and a mini-lesson on melodic structure, specifically of dotted rhythms and large melodic intervals, was possible. Imagine the different result if I had begun by teaching

Aaron Copland

melodic structure using the unfamiliar movie theme. The context would have been all wrong, and the results would have been disappointing. By the way, John Williams’ film scores (familiar from daily life) are an excellent connector to music by Aaron Copland (a varied context) whose use of perfect fifths has been called “The American Sound” in so far as American symphonic music is concerned.

The connections students make will inform the choices they make when they create, perform, and respond to music. Students who grow up listening to jazz will demonstrate this knowledge, interest, and experience in the rhythms, melodies and harmonies that show up in their musical creations, and in the “flavor” of their interpretation of musical works composed by others. It will also influence to what their ear is drawn when they are listening to music. That jazz-oriented students will likely hear the arpeggiated trombones at the end of Dvorak’s Symphony “from the New World” as a boogie-woogie riff, whereas someone not familiar with jazz will just hear the same passage as part of the exciting buildup at the end of that symphony (which is probably all it was intended to be).

I began this article talking about goals and objectives, and then have been discussing contexts and connections ever since. I would like to conclude by returning to goals and objectives, but now within a better context in which to understand them. There are at least two kinds of goals we should use with our students; these are academic goals and character goals. The specifics of each kind will need to wait for another post, but for our purposes here, I will use an academic goal for an example. There is something compelling about the Rondo known as “Fur Elise” by Beethoven. Middle school students seem to almost universally be drawn to it and many of them will work very hard to be able to at least play the first theme on the piano. It seems it has become a sort of rite of passage to learn this theme, and so it is passed on from student to student as they teach it to each other, or come to me to teach to them in small groups. Clearly, this bit of Beethoven is part of their daily lives and as such can be connected with various contexts which may include dedications (Fur Elise means for Elise, indicating that the piece was dedicated to someone named Elise, though exactly who is unclear). Fur Elise could then be part of a unit that included other works, perhaps in varied genres, that were also known to be dedicated to individuals. Such a unit would establish one of the purposes for which music is sometimes written. The student may begin with a straight forward performance objective. “I want to be able to play the first theme from ‘Fur Elise’ by Beethoven.”  Doing so may involve reading music, especially if the music teacher is teaching the student to play the theme. If so, another objective might be, “I want to improve my music reading so that I can play the first 32 measures of ‘Moonlight’ sonata by Beethoven in time for my sister’s birthday in November.” Notice now the objective has a something in which the student will demonstrate growth (improve my music reading,” something the student will be able to do as a result of instruction and which will demonstrate the desired growth (play the first 32 measures of “Moonlight Sonata” and a time in which the objective will be completed (“in time for my sister’s birthday in November”). This instruction will be packed with relevance and connections for this student, and so is an excellent example of writing an objective around connections.

# Artful Learning

In this, the centennial anniversary of Leonard Bernstein’s birth, all sorts of things that this American musical icon did are being brought out into the public consciousness. Of course, most know of Bernstein’s work as a conductor, composer and teacher, what with his numerous recordings, lectures, young peoples concerts and musical compositions. Of all that he accomplished, he once said that he was most proud of those young peoples concerts. He was so devoted to teaching, that he developed what he hoped would become a school reform method called Artful Learning. This is a method I have just begun to look at, but it intrigues me enough to want to share with you, and to begin using at least its principles to benefit my students.

Bernstein often referred to “universality” in reference to music. He believed that the expressiveness and enjoyment of music was universal, but he also understood that teaching was necessary in order to enable people to fully enjoy and fully appreciate the music he so loved to conduct, talk about, and perform on the piano. Out of this belief system came Artful Learning. In a nutshell, the method is to begin with a concept; not a musical concept but a universal one (there’s that word again). For example, let’s take the concept of enculturation–the familiarizing of a population of people with cultural norms–theirs or another. Let’s propose an essential question regarding our concept of enculturation. How does enculturation make life more meaningful? That question will guide our entire instructional unit.

Now that we have a concept and an essential question, we need a masterwork that is relevant to them both and that will serve as the basis for student inquiry. Let’s select “The One And Only Cereal” from A Quiet Place by Leonard Bernstein. The masterwork doesn’t have to be a Bernstein work, nor does it have to be a musical work, but in our example we will use music. The students, with the essential question in mind, listen to the masterwork, and then respond to it. In this masterwork, there are at least three cultures represented–jazz with its prominently African American roots, Western European 19th century art music, with its prominently Anglo and European roots,  and serial music, with its prominently academic roots. Most listeners will not only hear the different cultures represented, but will be confused by some while easily taking in others. This is because listeners will have been enculturated in one or two but not in the others. The students can experience the differences in listening to music that makes sense compared to music that does not make sense due to either being enculturated into that musical culture or not. The students experience the masterwork in a multi sensory way, through not only listening to the music, but moving (or trying to move) to the music,  through drawing, interacting physically with objects, like tossing a scarf gracefully or wringing a towel aggressively, through making facial expressions to any number of other visual, auditory, or kinesthetic responses they or the teacher might propose. All of this is the first phase in the Artful Learning Sequence, and that stage is called Experience.

Throughout the experience stage, the students will collect observations and questions from their experience of the masterwork. These observations and questions will be the basis for the next phase, which is inquire. The essential question focuses the students inquiry. Students research the essential question in light of their experience of the masterwork,  and engage in hands-on learning tasks to test, probe, demonstrate and explain the concept, which you will recall presently is enculturation. Students research the concept not only from a musical standpoint, but also using the interdisciplinary content to investigate the subject matter even more deeply. This is key to the method. It is designed to be interdisciplinary. One of Bernstein’s favorite mantras was that  “the best way to know a thing is in the context of another discipline.” What was it like to arrive in a new country with no experience with that country’s culture, practices, language, or just general way of doing things? Students can then connect being a new immigrant in unfamiliar surroundings with being a new music listener experiencing unfamiliar musical surroundings. It takes time to learn your way around. What are some things a person new to our country would want to know right away to feel more comfortable and at home? What are some things you’d like to know to make you feel more comfortable with serial music, or jazz, or classical music? What musical idiom are you most familiar with and how would you help someone who had never heard your music before come to understand it and enjoy it? You see how the inquiry builds momentum as it goes along?

The third phase is create. Students use their learning and creative ideas to create an original work that manifests their understanding of the concept, which in our example is enculturation. This could be a play, ballet, song, poem, or whatever students and teacher can come up with. Students consider several possible mediums for their project to determine  how best to represent the academic content from their unit of study in an original artistic work. They first construct a rough draft of the work,  then continue to evaluate and revise their it until they determine that it is ready for presentation.

Once the original artistic work is ready for presentation, students are ready for the fourth and final phase, reflect. Students consider the process they have worked through metacognitively, asking themselves how they learned, and cognitively, asking themselves what they learned. They document these reflections with detailed narratives. Students discover connections consider practical applications of their new knowledge. All of this strengthens students as more self-directed as learners.

Much of what I have described here is not new to many music educators. We routinely teach about repertoire, creating cultural and purpose contexts for the musical works we teach, and we often bring ideas and concepts from other disciplines into our music lessons. Social studies (historical context), science (the science of sound), math (ratios of sound durations and beat groups) and Language Arts (dramatic form) all have been integrated into our music teaching for quite some time. What makes this method different is that it places artistic works front and center. It showcases not only the artistic excellence with which they were born, but of the universality of harmony, emotions, collaboration (of sounds in music and of performers in presenting it). It also makes the arts desirable and accessible to teacher other than of the arts; in fact, Artful Learning is intended to be a school-wide practice wherein teachers of all subjects use the arts to teach their discipline in this arts-centered interdisciplinary approach. I will be using this method initially to teach a unit on Latin American Music this coming school year. I will be writing periodically on my progress.

# Child Development and Music Education

Dr. James Comer of Yale University has found six pathways along which children develop. These pathways are described as physical, cognitive, language, social, ethical, and psychological. While music education clearly has ties to all six pathways, I would like to focus in on two of them: cognitive and psychological.

The Cognitive Pathway and Music

The cognitive pathway addresses critical and creative thinking, and applying learning to accomplishing goals. It encompasses the highest levels of cognitive activity on Bloom’s classic taxonomy, those of analyzing, evaluating, and synthesizing. Critical thinking is the  objective analysis of facts for the purpose of arriving at an unbiased and informed conclusion. Creative thinking generates an element of newness into an observed entity. Creative thinking may result in a new product or artistic work, an new insight or interpretation of a pre-existing object, or a new approach to or way of thinking about something.

Anyone who is preparing to perform a musical work engages in both critical and creative thinking. Critical thinking is involved in selecting, analyzing, rehearsing and refining, and determining when a performance is ready to present to an audience. When selecting music to perform, musicians consider there own knowledge of musical works, understanding of their own technical skill, and the context in which the work is to be performed. Each of these considerations requires critical thinking. The musician must evaluate his or her own knowledge, reflect and assess their own technical skill as compared to the technical skill that will be required to perform the work,  ascertained through analysis, and the appropriateness to the anticipated audience and physical surroundings that is anticipated at the performance. All of this must be synthesized into a final judgment as to the merits of performing the particular musical work.  Once a work is selected, further types of analyses need to made on the music including harmonic, thematic, structural, expressive intent, and so forth. Once the rehearsals begin, the musician is constantly evaluating what he or she has just done, and planning what improvements and corrections need to be implemented during the next attempt. This is a cyclical process that continues until the rehearsal process is completed. Though many times the end of the rehearsal process is marked by a deadline, ideally, it should be ended when the performer(s) have evaluated their work and determined that the performance is ready to present to an audience. Throughout the process, there is abundant critical thinking being brought to bear.

Preparing a musical performance is not all about critical thinking, though. There is also the interpretive aspect of preparation. I mentioned in passing analyzing a musical work for expressive intent. This is an area in which critical thinking is of limited value because there is an absence of facts on which analysis can be performed. A listener’s interpretation of music can be influenced by non-musical factors such as life experience

Events + Thoughts = Feelings

and associations, and prior knowledge about the composer. For example, Margulis, Levine, Simchy-Gross, and Kroger (2017) found that when listeners were given positive information about a composer they were more likely to hear their music as happy, whereas when they were given negative information about the composer, they were more likely to hear the music as sad. A person’s own emotional status, especially with younger children, can also be transferred to music they hear, independent of a composer’s intent. Consequently, analyzing expressive intent cannot be done with “cold hard facts,” but instead with clues the composer leaves in the form of expressive elements and terms. Elements such as dissonance, accelerando, and crescendo tend to build tension, whereas resolution of dissonance, ritardando, and decrescendo tend to release tension. Low pitch can sound gloomy or scary, while mid-range pitches can sound relaxed. Isolated high pitches or low pitches can sound comical, while a low minor sonority can sound fatal or tragic. These are culturally normed emotional references that composers use and to which listeners respond with their imaginations and creative thinking. Still, they are only clues, and it is the purpose of a performer’s interpretation to convey the desired intent. That interpretation is arrived at, and rehearsed prior to presentation with the use of creative thinking.

The Psychological Pathway and Music

The psychological pathway is about an individual’s self-image and self-esteem. It includes their concept of self worth and competence, and ability to appropriately manage emotions. Research into the relationship between self-image and musical experience has been inconclusive. Whereas success in musical activities does tend to raise self image of musical ability, it does not necessarily raise self image in general. Music has been shown to be an effective aid in altering or controlling emotions. People often use music to reinforce a pleasant emotion that are experiencing, or to change an undesirable emotion that want to change. One of the  strongest foundations for advocating for music education is that music provides a healthy outlet for emotional expression. Just as students can use their language pathway to resolve conflict with words instead of violence, they can use their psychological pathway to control negative emotions by engaging with music.

According to researchers, there are several ways we listen to music in order to better manage our emotions:

• Entertainment – listening to music to maintain a positive mood or to evoke positive emotions.
• Revival – listening to music to relax or get energized.
• Diversion – listening to music to forget about something undesirable.
• Discharge – listening to music to release an emotion, such as anger.
• Strong Sensation – listening to music to stimulate our senses in new ways.
• Mental work – listening to music to get inspired or get new ideas.
• Solace – listening to music to experience comfort after an unfortunate event.

These are all examples of the different ways we may listen to music in order to regulate our emotions and channel them in positive ways. In research from Gothenburg University, listening to music was one of the most frequently reported main activities. Of the music-related experiences, up to 67% of individuals reported that listening to music had changed their emotions. Most of these emotions were reported to change in positive ways. These changes were most reliable when the music used was of the listener’s own choosing, compared to music that someone else (a music teacher, perhaps) chose for them. This last point highlights the importance not only of music in managing emotions, but in allowing students to select music not only to perform, but also to which to listen.

Music has a legitimate and important place in the physical development of children. Its emotionally charged and expressive nature, the manner in which it is performed and heard in communities, and the ways it engages the physical, cognitive, emotional, and psychological dimensions of humanity are proof positive that music is key to healthy human development.

Margulis, E. H., Levine, W. H., Simchy-Gross, R., & Kroger, C. (January 01, 2017). Expressive intent, ambiguity, and aesthetic experiences of music and poetry. Plos One, 12, 7.)

# Can rhythms be fast?

Tempo is a deceptively tricky musical concept. On the face of it, it seems straight forward enough. Tempo is measured as the number of beats occurring in one minute given a steady rate, and that beat can be equal to any note duration, such as eighth, quarter, half, or whole note. There are tempo markings that broadly indicate that the tempo should be lively, very fast, moderately fast, moderate, slow, or very slow. There are the more precise metronome markings that indicate a precise number of beats per minute, and the note value that will be used as the unit of measurement. All of this makes tempo uncomplicated and clear to performers, because as the musician plays or sings, they are forming rhythms over a concrete pace of pulses that coincides with the instruction in the printed score or, as in the case of dance music, of the standard convention.

Tempo for the listener is more complicated. The listener does not necessarily know what the unit of pulse is, and so must match a pulse rate with the rhythm patterns they are perceiving. So while the performer may be playing a passage of 32nd notes at a slow 8th note tempo (a common situation in classical slow concerto or sonata movements), those 32nd notes are going by rapidly for the listener, who might organize the music into beats of 16th notes, making the tempo faster than for the performer, who is measuring those same 32nd notes in slowly moving 8th notes. In this case, it would be tempting to say that the tempo (measured in 8th notes) is slow, while the rhythm (as perceived by the listener) is fast. But the difference is not between tempo and rhythm, but instead between the unit used to measure (and perceive) rhythm. The same music can be said to be fast or slow depending on what note value is being used as the unit of the pulse.

A good example of this is the opening of the 4th movement of Mozart’s Symphony no. 41 in C major (Jupiter). The tempo marking is motto allegro, and the pulse is generally around 120 beats per minute. Yet the first four measures are whole notes, and so one note progresses to the next slowly, even as one perceives the pulse to be fast, in contrast to the accompanying eighth notes, which are flying four times faster than the fast pulse, almost to fast to track. Yet if we listen to the same music and track two measures as one beat, though many notes pass by, the tempo now seems extremely slow. It is all in what is perceived as the unit of pulse.

A second factor in the perception of tempo is meter. Meter is part of the rhythmic structure of music, and influences how listeners perceive the unit of pulse. In the Mozart example just cited, the tempo is only perceived as fast if the meter is perceived as alla breve. If the meter were perceived as two or four whole notes per measure, then the tempo is perceived as Andante at most. Meter defines how the listener groups note durations into patterns that can be divided and subdivided into equal parts. There are times when musicians will use a faster, subdivided tempo to improve accuracy, while they intend the audience to perceive a slower, unsubdivided tempo. The introduction to Dvorak’s symphony no. 9 (From the New World) comes to mind. Notice how the conductor conducts eighth notes an an Allegro tempo, while the music, when listen to without following the conductor, is perceived as being Adagio, as Dvorak intended.

I began by asking the question, “can rhythms be fast?” We are now in a position to answer that question by saying no, it cannot. The reason is that tempo is a measurement of degrees of fastness measured in beats per minute, whereas rhythms are a relationship between a beat and a duration which is shorter, equal to, or longer than one beat. Rhythms as they are perceived by a listener are not individual notes, but patterns of note durations perceived as patterns by their relationship to a beat, regardless of tempo. In other words, the rhythm pattern of one quarter note, two eighth notes, two eighth notes again, and  one more quarter note will be heard as such at any tempo as long as the quarter note is used as the unit of pulse. The notes can be made faster by increasing the tempo. The first sound is equal to a beat, the next four sounds are divisions of the beat into two equal parts, and the last sound is again equal to a beat. We cannot say the rhythm is faster or slower, because the fastness or slowness is entirely dependent on the tempo, the speed of the beats, not the durations, which set the interval of time from the end of one note to the beginning of the next.

While it is true that we arrive at the next note sooner if the last note was a sixteenth note than if it were a quarter note, the reason we arrive sooner is a shorter note duration, not a faster tempo. The tempo, which is the measurement of fastness, has not increased, the durations of notes, the measure of rhythm, has decreased. There is more activity within the beat divided into four equal parts than within the beat divided into two equal parts, but that is not an indicator of faster, of tempo, but of duration, of rhythm. Fast does not exist apart from a reference to pulse. Fast is a relative concept that is not dependent on duration, but on pulse. A flourish of 32nd notes is a group of very short durations, not very fast notes. Notes are not fast or slow apart from the pulse to which they are sounded, only the pulse itself can be considered fast or slow.

# Toccata Blocks: A Tool To Help Teach Rhythm

No matter what method you use to teach music, be it Kodaly, Orff, or any other, when it comes to music reading there are certain aspects of our music notational system that are counter-intuitive and confusing to students who are just beginning. One of those difficulties is often the irrelevance of how the notes are spaced on the page. Students naturally assume that notes that are closer together go faster, and notes that are spaced further apart go slower. They will even carry this into the same note value. For example, they will think that quarter notes spaced closely together go faster than quarter notes spaced further apart. The concept of how the note head, stem, and/or beam are drawn can become overlooked, leading the student to make frequent rhythmic errors.

It is always good pedagogy to start from where students are and work from there to where you want them to arrive. Rather than dismiss using spacial perception as wrong, why not take advantage of children’s intuitive ability to perceive spacing in teaching them to read note durations accurately? Catherine Schane-Lydon has invented Toccata Blocks that do just that.

The basic set includes blocks with time signatures and notes on blocks that fit onto a provided easel. The easel is exactly the width of the blocks for a time signature and the correct number of beats of notes. A whole note block is the width of four quarter note blocks or four paired eighth note blocks. Single eighth note blocks are half the width of quarter note blocks and so forth. There are easels for simple and compound time signatures. Once an easel and time signature are chosen, the child builds a measure of rhythm by placing blocks on the easel. If the easel is less than filled, the child knows more notes are needed. If the easel is over filled, with a block hanging over the end, the child knows there are too many beats. This design makes building rhythms self-correcting because the child knows when he or she has done it right because the blocks will exactly fit across the easel.

I gave a set to students in middle school, and students in 2nd grade. An 8th grader said, “it [toccata blocks] helped me learn how the notes go together.” A 2nd grader remarked, “making rhythms is fun with these blocks. I got it wrong at first, but now it fits.”

I also used my set of Toccata Blocks to do a full class demonstration. I began a rhythm and then called on student to finish the measure. After each addition of a block, I asked the class if the measure was finished. “Do the blocks fit perfectly?” The children would look and respond, then give me suggestions on what block I should put in next. If the next block hung over the end, they were quick to reject that choice and make another of a shorter note duration. Almost every student I gave these blocks to to use immediately understood how they worked, and were able to correctly create a measure of rhythms.

Of course, I want students to write rhythms on conventional music paper, so it was important for me to make sure they took note of what notes they were using, and didn’t just fill up the easel randomly. So I had them tell me with each block they added how many beats that block added, how many beats they had, and how many more they needed. It was helpful to them to learn the note durations, and it delighted our school math coach!

Once they wrote original rhythms on paper, they could go back to the Toccata Blocks to check their work. They would exactly place what they had written on the easel, and see if it properly filled the easel or not. If not, they could not make corrections with the Toccata Blocks, they had to make corrections on their paper, and then return to the Toccata Blocks to again check their work.

With the length of the blocks proportional to the duration of the note, it is also possible to use the blocks as prompts for creative movement. The whole note block is long, and so a long, extended movement is called for. The quarter note blocks are short, so they call for smaller movements. Turning this into a movement game, children draw blocks like playing cards, then begin walking around the room. If a whole note is drawn, one giant step that takes four beats is taken. If a quarter note is drawn, then four smaller steps that traverse the same distance as the child who took the giant step is taken. Using them in this way helps students deepen their understanding of why the blocks are different lengths, and how the various note durations relate to each other.

Students worked well in small groups finding blocks to add to a group composed rhythm. I have learning centers set up in my classroom, and the Toccata Blocks make a good basis for such a center. With my older students, students who have composed a rhythm on the Toccata Blocks can then take it to students working on the keyboard to add pitches to the rhythm, making a short, one-measure melody. This rhythm then can become the basis for extending a melody.

The blocks are made of durable hard plastic. I anticipate that they will stand up well to classroom use. There is also a CD included with rhythms for students to build with the Toccata Blocks, taking dictation from the CD. The basic set includes quarter and eighth note durations in simple and compound time signatures. Supplemental sets add sixteenth notes to the basic set. Each block has a note on one side and the equivalent rest on the reverse side, so students can learn both notes and rests together. Toccata blocks are suitable for students in second grade and older, though one must be aware that many of the blocks could be a choking hazard for children prone to put such things in their mouth. I have had encouraging success with Toccata Blocks. They are certainly worth looking into. For more information, go to toccatablocks.com.

# Working from an Objective to a Lesson Plan

Let’s say you want your children to pass an object in time to the beat around a circle while chanting a rhyme to that beat with the correct rhythms. There are several competencies enfolded into that one objective. You want your children to be able to pass an object around a circle, you want them to pass on a beat and receive on a beat, you want them to chant rhythms accurately, and you want them to keep a steady beat with their chanting, and you want them to keep a steady beat with their movements. The fact is, playing a “simple” object passing game is not so simple after all. Let’s break this activity down into a sequence of learning activities that will prepare those children to succeed at playing the game.

First, we need a rhyme.

Just a side note for this is that when I play games that exclude a player, I always give them something to do once they are tagged “out.” For this game, I would have them leave the circle and take up a rhythm instrument to play either the beat or the rhythm while the circle continued to chant and play the game. Now back to our lesson plan.

There are two rhythm patterns in this chant; three quarter notes followed by a quarter rest, and four quarter notes. The first thing to do is to get all the children keeping a steady beat. I don’t want them making sounds at this point, because I want them to be able to focus on hearing me chant to them and hearing themselves chant back to me. So I will have them do a silent time keeping movement. Tapping the back of their left hand with the fingertips of their right hand works well. With the children doing this beat in this way, I will have them listen to me chant one of the patterns on a neutral syllable, and then have them repeat back to me what they just heard me do. Bum, bum, bum, — . They repeat, bum, bum, bum — . I would do this pattern at least twice, with everyone chanting together. Then I would do the other pattern. Bum, bum, bum, bum. They repeat, bum, bum, bum, bum. Again I would do this at least twice with everyone chanting together, then doing sometimes one pattern, other times the other pattern. Finally, for this step, I would have individual children chant one or the other pattern, still repeating it after me. Remember, all the time, the children are tapping the back of their left hand with the fingertips of their right hand.

By using a neutral syllable, I have helped the children focus on the rhythm without being distracted by the words. Now that they have learned the rhythm, I would now replace the neutral syllables with the words of the chant. I am not using rhythm syllables here because I don’t want the children to associate the rhythms with both syllables and lyrics in a new song all at once. Today, I need them to play the game with the words. Another time, I will introduce the rhythm syllables to replace the neutral syllables, and then go back to the words, which by then will be familiar, to play the game again.

So now the children are chanting the rhyme and tapping the beat. The next thing we need them to do is to pass an object on the beat; that is, to pass an object at the same time they are presently tapping. This can be a challenge, especially for PK and K students, so some readiness may be necessary. They already have their left hands held out in front of them for tapping with their right fingertips. Now they are going to right tap the left hand of the child to their right on one beat, and return their right hand to tap their own left hand on the next beat. I call this passing a beat. To practice this, the children temporarily leave off chanting the rhyme and instead chant “pass, own.” “Pass” refers to their neighbor’s hand, while “own” refers to their own left hand. This is the motion they will use to pass the object. When they can do “pass-own” well, have them start passing an actual object. They will continue to do “pass-own” throughout, but when the object comes to them, they will using the “pass-own” motion to actually pass the object.

Finally, have them continue to do the “pass-own” motion and to pass the object when it comes to them, but now they will chant the words of the rhyme instead of saying “pass-own.” Another teacher has had success assigning each student a number. The children count out loud from one through whatever number is assigned to the last child. The children are to pass the object to the friend whose number is being chanted when it is chanted. If this method is used, then the children only count on the “pass” motion and never on the “own” motion.

With younger children, you will need to repeat the readiness activity described above, though more briefly than at first, before playing the game outright. Eventually, the class will be able to sit in a circle and play the game straight away. At that point, it probably becomes a favorite activity, and so is best placed at the beginning or the end of the lesson. Because it was brand new in our hypothetical lesson, I would have placed it in the middle of the lesson, making it the most “meaty” segment of the class. Once the song is familiar, and the children can sing it without assistance, and play the game with no review, then it can be used outside the game for the literacy segment of the lesson plan, which is what I referred to as “meaty” above, the middle segment.

This middle segment is where I would start to use the rhythm syllables instead of neutral syllables. I would follow Feierabend’s Conversational Solfege procedure. The first step, teach by rote with a neutral syllable, as was done above. Next, teach the same material by rote with rhythm syllables. This might be during the same class, but often will be at a subsequent meeting. After that, have students decode; you sing the rhythm patterns in neutral syllables, and they sing the same patterns back to you in rhythm syllables. Again, decoding would not be done the same day they learn the syllables for the first time. I always wait until they have gained proficiency at one step before moving on to the next with a particular song or chant.

When the children can decode, it demonstrates that they have succeeded in associating the sounds of chanted rhythms with the names of each sound within the rhythms. Once they have decoded, then they can read what they have decoded. You chant, then they chant, while you point to the notated rhythms on the board. Now they are associating the melded sounds/names with the visual notation. After that, the students will be able to read the rhythms with rhythm syllables off the board without you having to chant it to them first. Do this with familiar songs and chants first, then with unfamiliar songs and chants to see if they can generalize what they have learned to new material. This is all done in the middle segment of your lesson plan, though not all in the same lesson.

The final third of the lesson returns to something lighter and something the children enjoy doing. I like to do my response to listening here. I use music they enjoy, and give them specific things to listen for and respond to. For example, I might ask them to tell me how the composer used timbre to create the image of water fountains (Respighi’s Fountains of Rome) or how what effect was created by changes in dynamics in Mozart’s Overture to The Marriage of Figaro. Or, I might play two popular songs, and ask them to name one thing in each that interested them and give one reason why. Or, I might play something twice, once to gather ideas and the second to create movement for expression, or for form. So the overall form of my lesson are hands-on music-making, music literacy, responding/connecting.

# More On Bloom’s Revised Taxonomy

In order to work effectively with Bloom’s Revised Taxonomy, we must understand two dimensions of learning: cognitive process, and knowledge. Cognitive process describes what thought task a learner is performing on a given text or focus. These include, in order of complexity from simple to complex,  remembering, understanding, applying, analyzing, evaluating, and creating. Of these, understanding is the most broad and most easily misused.

Understanding answers the question, “what does it mean?” It can involve interpreting, explaining, predicting, or comparing. Understanding is essentially knowing what an author, composer, songwriter, or visual artist intended to convey through their work, and being sufficiently familiar with a creator’s genre to predict what might come next in a musical or literary work, or what a visual artist’s next work might look like based on trends and characteristics found in recent works. Students also demonstrate understanding through comparing when they select a work they prefer from two or more, and then are able to explain why they prefer that one work over the others. Understanding is essentially thinking about artistic work, and making connections with other idioms, cultures, and other artistic works. As such, understanding is an intellectual endeavor.

Applying puts what has been learned through intellectual exercise to practical use. If through interpreting a musical work a student has learned how a composer typically uses specific musical elements and to what expressive purpose, than the student can use that knowledge in using those musical elements in the same way when preparing a performance of that or another of the same composer’s work, or in determining the expressive intent of the same composer in another, perhaps unfamiliar work. Another example might be if, at the remembering level, a student has learned the definitions of several musical terms, that knowledge can be applied when those words are knowledgeably used in the course of writing about a composer’s musical work to which the student has just listened, or which the student is preparing for a performance. It is at the applying level that authentic assessments are found. These are assessment that require students to be evaluated on doing something that musicians actually do in the “real world,” rather than something that only students are asked to do and then do not do once they become working musicians or practicing amateur musicians.

These first three, remembering, understanding, and applying, make a convenient and for many teachers a comfortable learning sequence. Breaking out of these three domains can be challenging for teachers and students alike, but it is at the next three domains that the most rigorous instruction and learning take place. Analyzing answers the question “how does it work?”  Many people go through life enjoying music, able to understand what they are hearing, and apply what they know to everything from singing “Happy Birthday” to their children, to reflecting with a friend or spouse on a concert they both just attended together. But to get to the “next level,” a person unpacks the sequences of musical events they heard and looks into learning “how did he do that?” We might learn that we became suddenly tense and uneasy because the chords started on a progression that brought them further and further away from the tonic, or because the pedal tone became more and more dissonant over the chords above. Whereas through remembering, students might identify what a composer did, (he used a crescendo), through analysis, students can explain how a composer used a crescendo, or for what purpose.

Evaluating is more complex. Evaluation is assigning worth or value to an artistic work, or to a performance of an artistic work. To do so requires that the student first have some criteria for judging the quality of artistic work that can be used objectively on any artistic work at least within a given genre if not universally. In other words, the student must be able to know concretely what bad art looks or sounds like, and what good art looks and sounds like, and then must be able to identify what in the artist’s execution of the elements of his or her art was done well and what was done poorly. Often, the most difficult part of evaluating is agreeing on what criteria are to be used.

Many music teachers I talk to are surprised to find creating at the highest, most complex location on the taxonomy. It is also the most advanced step in Feierabend’s twelve step learning sequence for Conversational Solfege. The original Bloom’s taxonomy had synthesis at this location. Creating requires one to pull together everything one knows and can do, and pour it into something new and original. A person simply cannot create an artistic work if they cannot understand, apply what is understood, and evaluate artistic works of others. Students need to acquire command of the elements of their art, of how artists used them to convey an expressive intent, and what particular uses of them create the very best result. Lacking this foundation, students will not have the materials they need to work with, nor will they know what to do with them, or even if they have succeeded once they have, in some way, perhaps randomly, put them together to form a work which lacks expression or quality.

Even an improvisor must have experiences with hearing, generating, selecting, and sequencing sounds and combinations of sounds before he or she can successfully improvise a melody that makes rhythmic and tonal sense. Improvising a melody on an Off instrument with all but the pentatonic scale tones removed is not an act of creating, because no understanding, applying, analyzing or evaluating is needed. The child merely needs to remember to strike any tone bar to a pulse and rhythm pattern. They may be improvising the rhythm, as if they were playing a drum, but they are not improvising a melody, though one incidentally results from their remembering domain activity.

These 6 cognitive tasks, remembering, understanding, applying, analyzing, evaluating, and creating, are performed on knowledge. This knowledge also has domains. There are four domains of knowledge: factual, conceptual, procedural, and metacognitive. Factual knowledge matches up nicely with remembering, conceptual and procedural knowledge goes well with understanding and applying, and metacognitive knowledge is useful with analyzing and evaluating, because understanding how one things, how one’s cognitive process is used helps a student select and use an analytical or evaluative strategy. It should not be surprising that higher level knowledge requires higher level thinking. If we want our students to engage in the higher domains of the cognitive taxonomy, then we can help them by giving them knowledge to work with that demands higher level thinking. For example, when assigning students to analyze a musical work, first have them plan out how they will go about doing their analysis. This requires them to have or develop criteria for determining when to use appropriate procedures. Of course, this also requires that you have taught them appropriate procedures from which they are now qualified to choose the best ones for the task at hand. Then, once they have determined what procedures they will use, they must apply that procedural knowledge, along with perhaps factual and conceptual knowledge,  to actually completing the assignment. Because this is a taxonomy of knowledge domains, students must be proficient with the lower levels of knowledge before being asked to work with a higher level of knowledge. For example, if students are struggling with  conceptual knowledge, teach at that level to raise their proficiency before asking them to use procedural knowledge.

Effective use of Bloom’s Revised Taxonomy is an effective way to control the level of challenge and rigor teachers present to their students. The taxonomy should not be seen as a description of learning styles, because one cannot say that, for example, one student is very analytic and so should always be given analysis tasks. No, that student will fare no better on analysis if remembering and/or understanding is deficient. Whereas some students will struggle to reach the most complex domain, none can afford to skip a domain to get to a higher one.