An Interesting View on Classroom Management

The following was a question within the class: How do classroom management and discipline relate to or influence each other? Explain.

I consider the primary difference to be proactive versus reactive. Proactive measures reduce the need for reactive measures (although it's a good question if the former can ever completely relieve the need for the latter). Thus, if the procedures are clear to the students, and if the teacher is constantly engaging the students, then the need to correct problems will decrease. Small corrections can be easily applied, and they will look like proactive engagement instead of reactive corrections. Proactively managing the class means the teacher does not need to overpower the class with super-strict discipline. He/she only needs to guide the class with purposeful engagement.


My mentor teacher likened the process to riding and raising horses. Horses are stronger than humans, as well as lazy and headstrong in natural personality. Given a preference, a horse would rather eat grass 24/7, whether the rider wants to go somewhere or not; Even a well-trained horse in the hands of an untrained rider will not obey, but instead, will stop to eat grass at every opportunity. Even within the herd, there must be a leader, but that leader is often challenged to ensure the most capable horse leads the herd. If the leader is not up to the task, another leader will emerge. So, we have a really close metaphorical parallel to life in a classroom.

Without a good proactive management plan, the class will simply do what they want, metaphorically stopping to munch the grass at every opportunity. Rather than productively trotting down the trail, the teacher must resort to constantly reminding each horse to get going, but in this case, it becomes an exercise in herding cats. Furthermore, the herd of students will test the leader and push boundaries until they find out where those boundaries are, and failing to find boundaries simply means a student will emerge as the class leader to supplant the teacher. It takes a strong will, good management practices, and a lack of tolerance for testing to manage the metaphorical herd. Teachers do not need to saw at the reins of a horse; nor can they beat the herd into submission. Instead, they actively engage all students proactively to control the herd.

Thus, we have a balance between managing proactively and reactively. The better the teacher is at proactive management dictates how much reactive management the teacher must use to maintain control, but there must never be a question about who is in control.

Expert blind spots, and an interesting model of understanding

Wiggins & McTighe describe the expert blind spot as a failure to think critically about foundational ideas. Once we accept an idea and begin to build on it, it becomes that much harder to change that idea. It's difficult to reach pipes that we have poured a concrete foundation around. But, I'd like to explore another aspect of the blind spot.


I ran into an interesting article called the Conscious Competence Model. Imagine that you have learned a skill, but do not remember learning that skill. For example, I learned to swim before I can remember - about 3 or 4 years old. So, I have developed a skill that I am unable to teach, because I don't remember learning that skill - I have only hazy vignette memories that far back. This is true, because I've tried to teach children how to swim. It's a little like trying to describe how to speak. I can demonstrate a sound, or demonstrate a couple swimming strokes. I can even move the person's arms in the water while holding them afloat. But, the couple times I've tried to teach a kid how to swim have been pretty dismal failures, mainly because I don't remember how I learned. But, instead of regressing back to conscious competence where we remember the learning process, what's needed is a 5th level of reflective competence, supporting the axiom that a teacher also learns. Chapman (2010) suggests we might even call this category "re-conscious competence."

So, one expert blind spot resides in unconscious competence. We become so used to doing a skill that we forget how to teach it. Furthermore, we assume that students begin in the conscious incompetence category (#2). This is not true, and it creates a disconnect in learning, because the student is not only unaware that they are unaware (unconscious incompetence), but actually resists the notion that there is more to learn. In a construct analogy, the student has built a roof on their building of earlier skills and begun to add air conditioners, vents, and shingles. Thus, we see resistance to the move from calculator math to algebra, because the student doesn't want to tear the roof off and start building again. For example, the student has learned four basic operations for whole numbers, fractions, and decimals, and believes a calculator will support them the rest of the way through life. They are completely (and truly) unaware of the main idea of secondary education math. In a sense, these students have climbed one ladder to a plateau, but are not even aware that there is another cliff, and another ladder. How can we begin to teach if the student is unaware that there's something to learn?

I would suggest that there is another circle connecting to unconscious competence (bottom right of illustration) which I would label as unconscious complacency, or perhaps a return to unconscious incompetence. This circle would represent those who have not kept up with changes, or who have become complacent in what they think they know. You're an expert driver, but you might still succumb to the danger of inattention because you are unconsciously competent. An expert skydiver forgets the fear of their first dive, and many lose the same sense of caution. Or, how many times have we heard some incompetent moron proclaim that they have twenty years of experience when any outside observer would agree that the moron is making things worse? This gradual reversion represents another expert blind spot, and a potential trap any of us can fall into without continuing education. This unconscious complacency is similar to that described by Wiggins & McTighe in our reading, where we write our assumptions in stone, build our construct around it, and then fail to challenge the foundation.

So, we have three expert blind spots.

1. We have a blind spot when we are not conscious of the learning process from the learner's point of view - a failure to move into reflexive or re-conscious competence.
2. We have a blind spot when we fail to realize that a student is not conscious of their incompetence, particularly if we fail to make the transition to reflexive or re-conscious competence ourselves.
3. And we have a blind spot if we fail to progress, where time and progress dictate that we gradually move back to unconscious incompetence through complacency.

Reference Chapman, A. (2010). Retrieved May 30, 2011 from http://www.businessballs.com/consciouscompetencelearningmodel.htm. Diagram courtesy of Will Taylor, Chair, Department of Homeopathic Medicine, National College of Natural Medicine, Portland, Oregon, USA, March 2007.

Classroom Management

How the teacher should act. In a word, teachers must act professionally:

• The primary purpose of a teacher is the ability to create within the student a desire to know more about any given subject.
• Teachers must subscribe to standards of improvement and collegiality.
• Teachers must care about their students, learning names and some favorite characteristic early, becoming fully aware of all students’ strengths and weaknesses throughout the year.
• Teachers should not attempt to befriend students on a personal level. Education is not a popularity contest, and attempts to be one of the students will only backfire into an unruly and unmanageable class.
• Teachers must explain what proper behavior is, and then expect that proper behavior throughout the year.
• Teachers must care about the subject they teach, presenting lessons in an engaging manner. If the teacher wants the lesson to be memorable, the teacher must be memorable, perhaps even a little eccentric.
• Teachers must activate prior knowledge, scaffold progress, and assist students in expanding their constructs, using an I do, we do, you do model.

How students are expected to behave. As a teacher, I expect students to:

• Enter the classroom and sit quickly when the bell rings.
• Use a mechanical pencil instead of a wooden pencil and pencil sharpener.
• Begin bell work when the bell rings. There will be some work on the board every day.
• Pass their papers sideways to a neighbor and participate in grading.
• Stop, turn, and face me when I activate the "I want your attention" procedure (Wong, 2004).
• Turn cell phones to silent and keep them put away. If I see or hear a cell phone between class bells, I will place it on my desk the first time, and then take it to the disciplinary office the next.
• A quiet classroom is not necessarily productive, and a noisy classroom is not necessarily a problem.
• Look up an answer instead of quitting when they don’t know.

What the classroom might look and feel like:

• A quiet classroom is not necessarily learning, and a noisy classroom is not necessarily disruptive. I want a classroom that is participative and engaged.
• Participate in Socratic discussion by raising their hands. I do not want a chosen few to engage by blurting out answers while the rest of the class coasts. I want students to compete to answer questions in class.
• There will be quotes and inspirational sayings posted on the walls.
• Desks will be side-by-side in pairs for partner work, and there will be room for the teacher to circulate through the class.
• The teacher’s desk will be in a front corner, with a podium and/or computer whiteboard station at the front of the class.
• There will be a shelf of fiction and non-fiction books related to math and science.
• The class will be rich with examples and what-if scenarios that require application and synthesis of ideas.

How the teacher helps students conduct themselves properly:

• Engaging the students is the first step towards maintaining discipline within a class, followed immediately by the expectation of proper behavior. 
• The teacher will post and review the rules at the beginning of the year. Students will be given an explicit choice between being bossed around or doing what they know they should be doing.
• The teacher will explicitly convey high expectations every day.
• The teacher will create routines or procedures in class which promote order and learning, and reduce confusion about what to expect.
• The teacher will admit mistakes and correct them when necessary.

What the teacher should do about misbehavior:

• While I do not have any particular department policies at hand, my general inclinations towards misbehavior are as follows:
• Discipline is not about win-lose. Management is about win-win, and the purpose of discipline is to return an escalating situation back to the win-win ideals of management. Winning means the student succeeded.
• The teacher will not allow a misbehaving student to engage in argument about a punishment. This is not some idea of a trial with prosecution and defense.
• If reactive measures are needed, the student’s name goes on the board after one verbal cue. This represents one 5 minute detention prior to lunch, making that student nearly last in the lunch line. A second mark represents a second 5 minute detention. Any additional marks cause a referral to the disciplinary office for a parent phone call.
• The teacher reserves the right to escalate punishment past detention to the disciplinary office for significant infractions such as (but not limited to) abusive language, defiance of authority, fighting, tobacco (or drugs or alcohol), etc.

How students should be taught what is expected of them:

• Class will begin with orientation, where I will provide and review a syllabus, a copy of the rules, a copy of class procedures, and a copy of disciplinary procedures.
• Per Wong's suggestion, we will institute classroom procedures by practicing them.
• Rules and procedures will be posted on the wall.
• Every lesson will explicitly describe why the lesson is necessary and what constitutes success.
• Students will be challenged with appropriate scaffolding to engage in reasoning that exceeds the minimal benchmark for success.

What Is Literacy?

The thing I'm finding very interesting here is that some think literacy refers to whether the student can read and understand literature. It's not. To me, the definition is more general and more simple: Literacy means the ability to read to learn - independently.

Another misconception is that if a student knows how to read, they can read to learn. I think this approach is at the core of what is failing in today's high schools.

Finally, there's a misconception that reading to learn works across content, so only the english teacher needs to teach the ability. Every subject has content specific vocabulary and often, content specific form. For this reason, reading to learn needs to be taught explicitly in each content from first grade to college, and I argue that this ability supercedes actual course content.

This idea comes under the analogy of teaching a person to fish, as opposed to giving them a fish. Reading to learn (learning to fish) teaches the process while providing food. Simply teaching content (eating a fish) means you put dinner on the table and called the children - that doesn't mean the children learn how to cook. Failure to realize this leaves a hole in a student's education that a future teacher must plug (or in yeat's analogy, a large pile of extra wood to burn). To really mix metaphors, too many teachers like this shoot education in the foot (the wood pile sits out too long, and gets wet. Wet wood takes more heat to start a fire. You get the idea.). And this, in a nutshell, is why otherwise curious students don't seem to care - who was it that said anything free is presumed to have little value? Perhaps we should shift a little from giving content away, and teach them how to obtain content.

Teaching Our Students How to Fish

The word expository originates from the word expound, which is defined as “to set forth or state in detail; to explain” (Random House, 2011). Explain is defined as “to make known in detail; to make clear the cause or reason” (Random House, 2011). This essay is an example of expository text. Therefore, expository text presents facts and explains information in detail in the form of lists, time lines or sequences, comparison or contrast, cause and effect, or problem and solution (GCU Lecture Series, n.d.). It has been said that students learn to read early, and then read to learn for the rest of their lives (Leach, Scarborough, & Rescorla, 2003), but this does not happen automatically. As a matter of fact, this transition seems to be failing as we increasingly spoonfeed information to our students as though education consisted of trying to fill a bucket instead of lighting a fire (Pychyl, T., 2008). An ability to comprehend expository text must be based on calculated instruction on how to use expository text in each content area. More accurately, teaching a student how to read and retain expository texts (in each content area) takes priority over the actual content of the reading: The content is simply an example for the process, and while we want the student to retain the examples, the process is actually the priority. We are not giving fish to students; We are teaching students how to fish, and hope that they are fed by the ones they catch as they learn to fish.

The transition from learning to read to reading to learn begins in late grade school, and continues through high school. Teachers begin to assign homework and in-class reading from textbooks, web sites, and class handouts, gradually increasing expectations that students absorb information from the expository text contained in each. But, according to Red Orbit (2006),

Many upper elementary grade teachers, regrettably, presume that their students have mastered the fundamentals of reading. Assuming that students have basic comprehension skills, the teacher may not provide explicit instruction in strategic practices that provide the foundation to good reading comprehension (para. 4).

By the end of eighth grade, Nevada English Language Arts (ELA) Standards (2008) require students to “read expository and persuasive texts to comprehend, interpret, and evaluate for specific purposes” (Standard 4.0). Amongst the many specific content indicators, students must be able to (Nevada ELA, 2008: Standard 4.0):

• Explain how language clarifies ideas and concepts.

• Describe how an author uses concrete examples to explain abstract ideas.

• Identify the main idea.

• Evaluate the impact of sequential and/or chronological order.

• Evaluate a cause and its effect on events and/or relationships.

• Evaluate a problem and its solution.

• Describe main idea based on evidence.

• Analyze the development of an author’s argument, viewpoint, and/or perspective.

From the listed indicators, it is very clear that the ability to read and understand expository text is of paramount importance by the end of eighth grade, with similar (but expanded indicators) in twelfth grade.

Students that are able to meet these standards enjoy a tremendous advantage over students that cannot. According to Reading Expository Prose (n.d.),

Expository text makes up the bulk of what we read. In school this is no different. Thus students need to know how such texts work, how they should prepare to read them, and what to do once they begin reading such texts (para. 1).

It should be clear that students who can read at middle school level are not necessarily proficient at comprehending expository text. So, what can we do to promote comprehension of expository text? There are several steps that rely on direct instruction, modeling, and practice, but this is not meant to be an inclusive list. For additional suggestions, I recommend the four dimensions of think-alouds discussed by Lapp, Fisher, & Grant (2008) on pages 380 – 382, as well as the suggestions from Reading Expository Prose (n.d.).

First, I disagree with the GCU Lecture Series (n.d.) that expository text “has a low incidence of ideas that activate prior knowledge” (para. 1). Although a construct may require a new branch, an imaginative teacher can always find some tree to anchor that branch. Activating prior knowledge before reading an expository text remains as essential as it is for any other lesson plan, and an inventive mind and directed class discussion can usually find some relation between a new idea and an old schema. Simply relating to an existing schema helps, but we can do more to activate prior knowledge. Students can scan the title, the table of contents, a synopsis, or even section captions to make a prediction about what they are about to read (Lapp et al., 2008). A look ahead at figures and tables (and their captions) within the text can assist, as well.

As described above, there are several types of expository text. Describing and looking for the signal words for that type of text (GCU Lecture Series, n.d.) cues the students about which type they will be working with, which is an essential second step to comprehending text. Understanding the structure of the text can be every bit as important as pre-scanning captions and titles to activate prior knowledge. This exercise can be combined with the creation of a graphic organizer, which serves as a prompt for students that are more spatially oriented learners. For example, if the text describes a sequence or process, a timeline or flowchart can be created. A Venn diagram (such as the one included below) can be used for comparison and contrast, where the student separates ideas from one or more texts that are different while creating a synthesis of ideas that are similar. A flow chart can be constructed for cause and effect, and a table of pros and cons can be constructed for problem/solution text. Teachers should model, and students should explicitly discuss which type of expository text they are starting to read, and begin to craft a graphic as they read. Returning to the idea of process, it is essential that students learn to take notes while reading, and graphic organizers create a scaffold for these notes. The mental act of separating ideas clarifies the text, and the act of writing adds a physical aspect for tactile learners. Even hi-lighting text while reading creates a process where the reader separates essential ideas from examples, clarification, and illustrations.

Unfamiliar vocabulary constitutes one of the greatest difficulties in reading expository text. A teacher can prepare a vocabulary list from the text that the students are likely to be unfamiliar with, and instruct the students to be on the watch for these words. Not only will the list serve to activate prior knowledge and hint about the subject to come, but it will also serve as a standard or bar which tells the students when they have succeeded in understanding the text. Teachers can stop and discuss vocabulary regularly, using Socratic discussion based on context, root words, and affixes to prompt students into developing their own paraphrased definitions (Gifford and Gore, 2010; Lapp et al., 2008).

One of the biggest faults in student reading is the failure of metacognition. Essentially, the student only thinks they understand the text after they have finished reading. Working under the assumption that a student does not know the idea if they cannot express the idea, summary think-alouds (Lapp et al., 2008) provide informal assessment of class understanding and class progress towards comprehension. Small chunks of expository text can be read aloud by the teacher or by a student, with frequent stops to discuss each chunk. Modeling paraphrasing, followed by class discussion to paraphrase a section, scaffolds the process of metacognition within Vygotsky’s zone of proximal development. While teacher modeling is essential, peer rephrasing offers a far higher return (Gifford and Gore, 2010). Involving the class in Socratic dialogue to extract meaning from the writing creates interest, first by soliciting opinions, and then by adopting a class consensus that shows those opinions are valued.

Key to comprehending expository writing is the ability to select the most important sentence or phrase of a particular paragraph. Again, using the think-aloud system described by Lapp et al. (2008), teachers can model reading the paragraph and, instead of summarizing, pick the most important feature from that paragraph. This exercise is more related to hi-lighting than summarizing. Although we must discourage students from hi-lighting (and annotating) school textbooks, hi-lighting essential text and annotation of text are critical self-study reading skills that should be explicitly taught using handouts that students are permitted to mark.

This list of methods to enhance the understanding of expository text is by no means complete, and I do not even argue that these are the best methods for any class or any expository text. Many articles have been written, and a couple have been cited in this text which expand on (and add to) the ideas presented in this discussion. Nevertheless, I consider these ideas most useful for my style of teaching within math or science content. First, we must activate prior knowledge. Next, we must identify the text structure, creating the bones of a graphic which we will flesh out as we read. Vocabulary is essential as both a goal and a means, and student rephrased vocabulary is more important than a dictionary definition. A student does not have cognition unless they have metacognition, and student consensus of meaning enhances student understanding. Finally, students must be explicitly taught how to mark up text and create notes as they read.

References


GCU Lecture Series (n.d.). Literary elements of expository texts. GCU Lecture Series 4.1;
SED-435. Retrieved April 13, 2011 from http://angel03.gcu.edu/section/default.asp?id=549975

Gifford, M. & Gore, S. (2010). The effects of focused academic vocabulary instruction on
underperforming math students. Retrieved 3/31/2011 from http://www.ascd.org/ASCD/pdf/Building%20Academic%20Vocabulary/academic_vocabulary_math_white_paper_web.pdf

Lapp, D., Fisher, D., & Grant, M. (2008). “You can read this text – I’ll show you how”: Interactive comprehension instruction. Journal of Adolescent and Adult Literacy, 51:5. Retrieved April 10, 2011 from http://web.ebscohost.com.library.gcu.edu:2048/ehost/pdfviewer/pdfviewer?sid=d7f5eb56-a938-49ee-9878-50058ff53be9%40sessionmgr115&vid=2&hid=106

Leach, J., Scarborough, H. & Rescorla, L. (2003). Late-emerging reading disabilities. Journal of educational psychology, 95, 211-224. Retrieved 3/6/2011 from http://www.apa.org/pubs/journals/releases/edu-952211.pdf

Nevada English Language Arts (2008). Achievement indicators for reading: Grade 8: Content standard 4.0. Retrieved April 13, 2011 from http://www.doe.nv.gov/Standards/EngLang/Eighth_Grade_Achievement_Indicators.pdf

Pychyl, T. (2008). Don’t delay; Understanding procrastination and how to achieve our goals. Psychology Today. Retrieved April 3, 2011 from http://www.psychologytoday.com/blog/dont-delay/200805/education-is-not-the-filling-pail-the-lighting-fire

Random House (2011). Explain. Retrieved April 13, 2011 from http://dictionary.reference.com/browse/explain

Random House (2011). Expound. Retrieved April 13, 2011 from
http://dictionary.reference.com/browse/expound

Reading Expository Prose. (n.d.) Retrieved April 10, 2011 from
http://www.englishcompanion.com/room82/readexpository.html

Red Orbit (2006). Self-regulated strategy development instruction for expository text comprehension. Retrieved April 13, 2011 from http://www.redorbit.com/news/education/424135/selfregulated_strategy_development_instruction_for_expository_text_comprehension/

Education Philosophy and Rationale

Mission Statement

The primary purpose of an educator should be the creation within the student of a desire to learn more. Using technology, an educator will impart a broad base of mathematical and scientific principles, the ability to write literately and persuasively, and the ability to critically consider issues of art, history, geography, politics, practicality, and social mores. An educator will not discriminate on the basis of gender, gender preference, race, creed, income, social status, disability, or any other physical or social characteristic.

Webb, Metha & Jordan (2010) define philosophy as “the study of the fundamental nature of knowledge, reality, and existence” (p. 50). From Webb et al (2010), there are three traditional philosophies and two twentieth century philosophies that apply to education: Idealism, realism, and neo-Thomism (also called theistic realism) have existed for centuries, while pragmatism and existentialism are relative newcomers from the early twentieth century. Each philosophy has a component of metaphysics, epistemology and axiology. Metaphysics considers the nature of reality, epistemology considers the nature of knowledge, and axiology considers the nature of values (Webb et al, 2010).

Naturally, the first question is axiologic in nature: What value does this discussion have? Why should we care about amorphous philosophic generalizations, and what value does an examination of such a generalized subject hold for teachers and education? Webb et al (2010) writes, “To teach without a firm understanding of one’s personal philosophy and philosophy of education would be analogous to painting a portrait without the rudimentary knowledge and skills of basic design, perspective, or human anatomy” (p. 50). Perhaps the metaphor can be stated another way: If a mechanic persists in turning the nut the wrong way to loosen it, the problem will never be solved. It is only through knowledge of how things work that we can make things work. Philosophy forms the foundation upon which ideologies are built, and ideologies form the construction that allows the teacher to test and use theories (Philosophy, ideology, and theory, n.d.). Without a firm foundation in philosophy, a teacher is unable to make a selection of appropriate ideologies or theories from which to operate, creating a career which drifts from rock to shoal at the mercy of the current, instead of demonstrating personal control toward a goal. For this reason, educators are well served by the consideration and construction of their philosophy. Knowing which direction to apply force will help loosen the nut.

Webb et al (2010) defines metaphysics as descriptive of the nature of reality, which is subdivided into ontology (the nature of existence) and cosmology (the origin of existence). As an aspiring math or science teacher, my metaphysical philosophy must be based on realism. Webb et al (2010) describes metaphysical realism by stating, “the universe exists whether or not the human mind perceives it. Matter is primary and is considered an independent reality,” and “the interaction of matter and form is governed not by God but by scientific, natural laws” (p.56). This principle can hardly be disputed in science or mathematics. Mathematically, definition and principle are intertwined in ontology such that a number, regardless of language or culture, represents a discrete value or an independent reality. Scientifically, certain laws are presented, based on our understanding of ontology and cosmology, and even though the perception of these laws can change with new information, realism proposes that there is an ultimate law which is true. It is true that our knowledge of laws was once far more faulty than it is now, and undoubtedly there is much more to know. Up until the 1930s, Scientists believed that protons, neutrons and electrons were the smallest building blocks of creation, but we now know that these small particles are made up of even smaller particles called quarks (Fundamental particles, 2009). Realism does not present that all things are known. Instead, realism simply presents that there is a truth which we can come to know. Nevertheless, there are certain laws which are known, and are unlikely to change. For example, the discovery of quarks did not invalidate the known behavior of atoms. Einstein’s theory of relativity enhanced our understanding of gravity, space, and time, but a person would be unwise to leap from a height while questioning Newton’s reality of gravity. No matter how strong their ideal, considerations of the knower would have far less impact than their body. This metaphysical realism rules out idealism, pragmatism and existentialism, since each considers reality to be relative to the knower (Webb et al, 2010). Neo-Thomism presents a different metaphysical consideration, though. Neo-Thomism, also known as theistic realism, embraces reality, but rather, postulates that reality was created by God (Webb et al, 2010). Metaphysically, I do not reject neo-Thomism, but there are epistemologic concerns that cause me to reject this as a teaching philosophy.

Epistemology describes the nature of knowledge, defining ways of knowing, including logic, intuition, deduction, observation, etc (Webb et al, 2010). In terms of science and mathematics, the priority of perception (including measurement), followed by inductive and deductive reasoning to extrapolate meaning, appeals to my sense of realism. It is on this basis that I must reject neo-Thomism, because Thomas Aquinas, the philosopher for whom the theory was named, created a hierarchy of epistemology that places revelatory knowledge above reasoning, and reasoning above observational perception (Webb et al, 2010). I consider this opposite of true knowledge, where perception drives reason, and reason confirms revelatory knowledge.

Axiology describes the nature of values, questioning purpose in terms of aesthetic and societal values, as well as social mores (Webb et al, 2010). The value of a realistic foundation in science and math can hardly be debated. Math, science, and engineering careers pay well and provide highly autonomous careers, but even without pursuing a career directly related to math and science, a firm grounding in those disciplines provides a foundation for critical thinking. Without an understanding of how the world works, critical consideration of an issue is bound to be far less effective. Nevertheless, realism encourages a sense of equation to learning, which suggests didactic delivery and rote learning. Nothing is relative, this is the way things are, and we expect the student to learn a series of principles. But thought processes do not follow scientific principles of cause and effect, and students are not equations. Ormrod (2008) writes, “Contrary to what many students think, rote learning is a slow and relatively ineffective way of storing declarative information in long-term memory” (p.204). So, although realism supports the subject material of math and science metaphysically and epistemologically, realism fails the student axiologically in the analytic division between the process of delivery within education and the goals of education. Frankena, Burbules & Raybeck (n.d.) create a philosophic distinction between the process of delivery (aims, methods, and effects) and the goal of education (desired skills, knowledge, and beliefs), writing:

• Some such normative theory of education is implied in every instance of educational endeavor, for whatever education is purposely engaged in, it explicitly or implicitly assumed that certain dispositions are desirable and that certain methods are to be used in acquiring or fostering them, and any view on such matters is a normative theory of philosophy of education (para. 7).

Philosophies of methodology and disposition retain axiologic value in our consideration of purpose, and should be considered separately as a means to furthering education. So far, we have discussed realism in terms of the subject matter (disposition), but Frankena et al (n.d.) describes axiologic value in various methods that create the dispositions that we value.

Psychological theories of behaviorism, cognitivism, constructivism and the theory of scaffolding success create an ideal of more efficient absorption of information and mastery learning, but these theories are based on experimentalism (pragmatism), which we rejected metaphysically and epistemologically for lack of certainty in content. Nonetheless, Frankena et al (n.d.) creates a distinction that permits separate consideration of content and delivery. While realism forms the foundation for content, experimentalism forms the foundation for delivery, allowing the teacher to tailor the method of instruction for different learning abilities.

Regarding disposition, my philosophy is governed by realism, but there must be a distinction between disposition and method. Essentialism states there are fundamental objectives of non-debatable mathematic and scientific principles which students must know (Webb et al, 2010). Nevertheless, students are not equations, and do not behave or learn by principles of science. One overall philosophy will not work for disposition and method, and recognizing the value of behaviorism, cognitivism, constructivism, and scaffolding success requires the teacher to consider experimentalism in the delivery while maintaining realism for disposition.

Frankena, W.K., Burbules, N.C. & Raybeck, N. (n.d.). Philosophy of education: Historical overview, current trends. Retrieved August 5, 2010 from http://education.stateuniversity.com/pages/2321/Philosophy-Education.html


Fundamental particles (2009). Stanford Linear Accelerator Coalition (SLAC) National Accelerator Laboratory. Retrieved August 9, 2010 from http://www2.slac.stanford.edu/vvc/theory/fundamental.html


Ormrod, J. (2008). Educational psychology: Developing learners (6th ed.). Upper Saddle River,
NJ: Merrill Prentice Hall


Philosophy, ideology, and theory. (n.d.). EDU215 Education Foundations and Framework Module 2, Lecture Introduction. Phoenix, AZ: Grand Canyon University.


Webb, L.D., Metha, A. & Jordan, K.F. (2010). Foundations of American education (6th ed.). Upper Saddle River, NJ: Merrill

Secondary Math Instruction: The Primacy of Course Rationale

I attended classes with two math teachers in algebra 1, geometry, trigonometry, and calculus, as well as two physics classes in a secondary school in the Washoe County School District (Reno, Nevada). One thing I noticed was the difference in student motivation between the required lower level classes and the elective upper level classes. One could argue that students taking elective classes are more motivated than students taking required courses, but I disagree that this is the entire difference. A course rationale is important to create student motivation, and according to Ormrod (2008), intrinsic motivation is the most important factor in student success.

• Intrinsic motivation is ultimately what will sustain students over the long run. It will encourage them to make sense of and apply what they are studying and will increase the odds that they will continue to read and learn about writing, science, history, and other academic subject matter long after they have left their formal education behind (p. 386).

If we consider that an objective is important to a lesson or unit, then the same principle would apply to an objective for a course. To an already motivated student, a course rationale is self-evident and does not need to be explained. To the lower level (and struggling) students, mathematic drill and practice seems like busy work with little or no connection to job prospects after school. I asked my primary mentor how he motivated students, and his answer was, “they know this course is required to graduate.” I believe this rationale is insufficient to intrinsically motivate students to exceed standards and exacerbates the difference between advanced students that are already motivated and struggling students who fail to see the connection between school and work.

Mathematics is a theoretical discipline which builds foundational skills, but delays practical applications until they are used in science classes. For example, an equation is a statement of truth for a given situation, but what importance does a statement of truth have in a practice situation? And why do we need to practice? Why do we care if x equals 5 one time and 7 another time. Why do we use the letter x in some situations, y in others, and Greek letters in yet others? What is the practical application, and why do we need to know this? The answer is short and simple: In practice situations, we do not really care what the variable equals, but we want to know if the student can apply the principles of mathematics to real situations. To teach mathematics, we have found that repetition and practice cause success. Students must learn to crawl before they can walk, so there is a progression of muscle memory where the early material must become automatic before attempting advanced lessons. Therefore, we create practice problems for students that first, tests their ability to work a process (mathematics), and then teaches the student to apply that process to a practical situation (science). We conclude that the actual mathematics coursework teaches a student how to think. Unfortunately, intrinsic motivation is absent in a highly theoretical subject. Students see this as busy-work imposed by adults, missing the rationale that mathematics is essential to nearly every higher level career.

For example, consider the game of chess, which is an analogy of medieval war: There is no practical application to the game. A general will not win a battle by sitting in a tent over a board game, there is no king or queen in the United States, castles (rooks) and cavalry on horseback (knights) are anachronisms, bishops do not wield temporal power, and soldiers are not pawns to be discarded without blood. But, a general can win a battle by deciding what is important and what can be sacrificed, and by using assets to seize the initiative and force the actions of an enemy – all principles of a successful chess player. Chess teaches a tactician how to think. Similarly, mathematics teaches a student how to think, informing scientists and business people of the possible and the process – the tactics of deduction. Simply stated, without an understanding of the principles of mathematics through drill and practice, the graduate is unprepared for the business world. While advanced students who have chosen a college path are already motivated, course rationale is essential to freshmen and sophomore student motivation, and adult busy work creates a lack of motivation at best.

Bloom’s taxonomy (St. Edward’s University, 2001) and Wiggins & McTighe (1998) describe the assimilation of metaphysics and epistemology: What we know, and how we know it. More specifically, Bloom’s and Wiggins & McTighe describe constructive levels of knowledge, from the ability to repeat information to the ability to evaluate the accuracy of an answer. These distinctions are important to upper levels, but I argue that the first three levels of Bloom’s taxonomy (knowledge, comprehension, and application) are enough for secondary education math. Instead, mathematics must begin with the idea that there are simple rules to equations, and if the student applies those rules in simple processes, the student can succeed in moving from question to answer.

Implicit synthesis comes with practice and construction. Explicit synthesis and evaluation are more appropriate to baccalaureate studies. A grade school student learns that two plus two equals four, but number theory teaches a baccalaureate math major why. Side – angle – side is a geometric theorem, but Euclidian geometry creates the grounds for evaluation of that theorem. The upper levels of Wiggins & McTighe are even less relevant to the algebra 1 student. The idea that a student must have perspective, empathy, and self-knowledge for prejudices and opposing points of view in mathematics is nonsense. While these facets are essential in subjective courses, realism forms the underlying theory of mathematics. That doesn’t mean we allow our students to be contemptuous of a student with an incorrect answer (a lack of empathy for the fellow student), but it does mean that there is a correct and incorrect answer, and point of view does not affect the application of mathematics.

Therefore, realism and a vast depth of material require a student to learn to crawl first. A student must learn truths which provide the groundwork for mathematical proofs that can be debated in baccalaureate and graduate level mathematics. Synthesis occurs through a naturally constructive process inherent in mathematics as the student applies one lesson to the next, creating multiple levels of practice with one problem. For example, the first lesson below begins with the identity function for addition. The rest of the lessons use that identity function without re-teaching it, so synthesis occurs naturally through repeated application as we expand the construct. We have synthesis when algebra students can perform elementary math (addition, subtraction, multiplication, and division) on coefficients and exponents, and when algebra students can apply the rules of equation manipulation to calculus as they differentiate or integrate. Each advanced level reviews previous levels.

This is not to propose that Bloom’s or Wiggins & McTighe’s work is unimportant, but as a teacher, I would not explicitly attempt to teach these levels. On the other hand, I think it is essential that teachers provide – and students consider – the rationale of each course explicitly, beginning in junior high, and continuing implicitly through the rest of their education. Simply passing from one year to the next until graduation is insufficient rationale, which is insufficiently motivating for most students.

Teachers at the secondary education level begin their course with classroom procedures, course goals, grading system (rubric), and a syllabus. I want to create a discussion of larger goals in the next step, beginning with a discussion between the differences between a career and a job. How important is career satisfaction? What is autonomy? Let us get right down to the basics of money: How important is a good income, and how important is mathematics to a good income? There are 52 weeks in a year, and 40 hours in a week. Using 2000 hours per year as a reasonable rounding number, how much money does an adult need after taxes to pay rent (or mortgage), pay for health insurance, buy utilities (cable tv and internet with all the bells and whistles?), make car payments (BMW or Hyundai), buy car insurance, afford groceries (steak or hamburger), eventually marry and have children, and after all that, how much is left for luxuries and fun? Let us have a concrete discussion on realistic incomes versus realistic expenses for various professions in concrete terms of math. What is a good living? Is $20 per hour a lot or barely enough? How can a person in their young twenties earn that much or more? At this point, we can introduce the idea of a variable. Surprise; x and y are not just letters! They represent the numbers we discussed as income and expenses. Equations don’t represent busy work. Equations represent truths that define real-life situations from science to finance, and while we will practice with a lot of equations that don’t seem to be practical, this course is about a lot more than busy work! This process synthesizes previous math, allows the teacher to informally assess pre-algebraic abilities, creates a direction to proceed, and combined with the knowledge that mathematics is essential to virtually every profession, produces a motivation that the teacher can refer to through the rest of the year. The teacher is creating intrinsic motivation in their course, but as with other aspects of education, if following education does not review or refer to a previous lesson, the students will fail to apply that lesson. It becomes noise in a panoply of lessons, so the teacher must refer to motivation throughout the year.

The secondary education teacher has three primary purposes: They must create intrinsic motivation to learn the material. They must teach a student how to learn (neat note taking, how to retain textbook reading, simplifying problems and processes, synthesis of lessons, etc.). Finally, the teacher must provide the course content. Without the first two, the last one becomes drudgery with hit and miss results.

Unfortunately, I was not in a position to test this theory as a visitor in various classes, since this discussion is most proper as a foundation for each course. Nevertheless, I received a positive response to the idea from my primary mentor, whose primary concern involved care in denigrating parent’s jobs. I hope that students will take menial jobs in fast food and retail (no math required) to see how hard it is to work their way up the ladder to a position of responsibility, but upon graduation from high school, I want students to opt for a college or technical degree with substantial practical demand. Care must be taken not to denigrate the night shift assistant manager of a McDonald’s (there are usually two employees at night, so this is a grandiose title which entails menial work) as a means to motivation, in case someone’s parent has that job, but I want my students to aim much higher. That, and career tracks like it, are for students that fail to become intrinsically motivated.

Creation of student motivation is the single most important teaching skill. The course material, including the distinctions made by Bloom and Wiggins & McTighe, amount to little if the students cannot be motivated beyond the bar of repetition. In order to do this, the teacher must accomplish certain goals. First a higher bar must be created in the form of concrete income expectations, followed by continual reference to those goals throughout the course. Failure to participate in the rat race does not eliminate the maze - it only limits the rat’s options. Secondly, the coursework must seem possible. Far too many students think math is a difficult subject, and this illusion creates an expectation of negative reinforcement. Students attempt to avoid math if they think they are not adept, and the expectation becomes the reality. In reality, math is a subject with natural constructive continuity throughout the student’s education that sets the rules about what a person can do with numbers. A skillful teacher will make the coursework seem possible by simplifying the processes, providing scaffolding, building constructs, and reviewing backwards to create synthesis. Given appropriate classroom management techniques and a caring and engaging personality, a teacher that can accomplish these goals will be far more effective than the teacher that requires their course, offers no rationale, and fails to explicitly create a construct.

St. Edward’s University (2001). Blooms taxonomy wheel. Retrieved 1/20/2011 from http://www.in2edu.com/downloads/thinking/blooms_taxonomy_chart.pdf.


Nevada Department of Education (2007). Math achievement indicators. Retrieved 11/25/2010 from http://www.doe.nv.gov/Standards/Mathematics/Grade_12_Math_Achievement_Indicators.pdf.


Ormrod, J. E. (2008). Educational psychology: Developing learners, Sixth Edition. Merrill Prentice Hall. Pearson Education, Inc.


Wiggins, G., & McTighe, J. (1998). Understanding by design: The six facets of understanding. Retrieved 1/19/2011 from http://pdonline.ascd.org/pd_online/ubd_intro/wiggins98chapter4.html.