- 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).
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.
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