Making Mathematics Memorable

Posted on by Society for Classical Learning

This session will begin with a brief survey of works on the art of memory from those of the ancients to modern works on neuroplasticity. Then implications for the teaching of grammar school mathematics will be described, including the teaching of arithmetic proficiency through memorable concrete and visual experiences that result in enduring understanding.

Andrew Elizalde

Andrew Elizalde earned a B.A. degree at Depauw University where he graduated Phi Beta Kappa and Magna Cum Laude, earned a mathematics major, physics minor, and religious studies minor, and received the H.E.H. Greenleaf Award as the most outstanding graduate of the school’s mathematics program. His teaching experience includes work in both public and private schools as well as private tutoring in subjects ranging from elementary mathematics to advanced calculus and physics. Andrew currently serves as Dean of Academics, at Covenant Classical School in Fort Worth, Texas. Andrew also regularly o ers consulting services to schools striving to teach mathematics with a a distinctly classical and Christian framework and pedagogy. Andrew and his wife, Brooke, have three daughters.

Proof Writing Across the Curriculum

Posted on by Society for Classical Learning

The practice of proof writing is too often restricted to rigid two-column format and limited to the one-year experience know as upper school geometry. In this session I will make a case for teachng proof writing, the composing and presenting of convincing mathematical arguments, in a variety of formats across the mathematics curriculum. Then I will present the following series of proofs/mathematical arguments accessible to middle and upper school students: deriving the formula for the area of a circle; proving the infinitude of primes; two proofs of the Pythagorean Theorem; proving the irrationality of the square root of two; visualizing the derivation of the quadratic formula; and proving the Fundamental Theorem of Calculus.

Andrew Elizalde

Andrew Elizalde earned a B.A. degree at Depauw University where he graduated Phi Beta Kappa and Magna Cum Laude, earned a mathematics major, physics minor, and religious studies minor, and received the H.E.H. Greenleaf Award as the most outstanding graduate of the school’s mathematics program. At California State University Long Beach, he obtained a teaching credential with a professional-clear qualification for his coursework regarding exceptional children and technology integration. His teaching experience includes work in both public and private schools as well as private tutoring in subjects ranging from elementary mathematics to advanced calculus and physics. Andrew currently serves as Dean of Academics, at Covenant Classical School in Fort Worth, Texas. His recent work most notably includes a comprehensive reform of the Veritas School (Richmond, VA) mathematics program as well as a restructuring of faculty professional development. Andrew also regularly offers consulting services to schools striving to teach mathematics with a a distinctly classical and Christian framework and pedagogy. Andrew and his wife, Brooke, have three daughters.

14 Essential Elements of Classical Christian Mathematics Instruction

Posted on by Society for Classical Learning

Contextualize through storytelling; love your subject out loud; color your classroom with traditionas and routines; engage all of the senses as often as possible; contemplate truth, goodness, and beauty; present archetypes worthy of imitation; teach Socratically through problem solving; move from the concrete to the pictorial to the abstract; derive and demystify formulas; canonize arguments and experiments; translate theories into applications; understand the limits of science and the cycle of the scientific enterprise; engage great works and primary sources and motivate service through doctrine and doxology. Descriptions of these fourteen essential elements of classical Christian mathematics instruction will be presented accompanied by specific examples and corresponding book recommendations.

Andrew Elizalde

Andrew Elizalde earned a B.A. degree at Depauw University where he graduated Phi Beta Kappa and Magna Cum Laude, earned a mathematics major, physics minor, and religious studies minor, and received the H.E.H. Greenleaf Award as the most outstanding graduate of the school’s mathematics program. At California State University Long Beach, he obtained a teaching credential with a professional-clear quali cation for his coursework regarding exceptional children and technology integration. His teaching experience includes work in both public and private schools as well as private tutoring in subjects ranging from elementary mathematics to advanced calculus and physics. Andrew currently serves as Dean of Academics, at Covenant Classical School in Fort Worth, Texas. His recent work most notably includes a comprehensive reform of the Veritas School (Richmond, VA) mathematics program as well as a restructuring of faculty professional development. Andrew also regularly o ers consulting services to schools striving to teach mathematics with a a distinctly classical and Christian framework and pedagogy. Andrew and his wife, Brooke, have three daughters.

Understanding the Current Condition

Posted on by Society for Classical Learning

Since the launch of Sputnik in 1957 educational reform in the United States has been generally motivated by a desire to be internationally competitive. Unfortunately, cycles of reform largely characterized by document-based attempts to teacher-proof curricula through standardization and high-stakes assessment have proven largely ineffective. Recent international assessments measuring the mathematical aptitude of students continue to rank the United States below the majority of developed countries. Corresponding qualitative and quantitative studies have identified teaching practices contributing to this ongoing mediocrity. Let us now give attention to these practices so that we do not continue to repeat the same mistakes. Then let us consider those elements of mathematics that have been entirely missed due to our recent obsession with international competitiveness.

Teachers taking their cues from oversized textbooks are finding their pedagogy inevitably compromised. In an effort to increase marketability by satisfying not one but rather a multitude of published standards, the vast majority of textbooks contain far too many sections for any one teacher to “cover” over the course of an academic year. Making it to the last page of a textbook without skipping sections requires teachers to move at a breathless pace, too often achieved by limiting instruction to a series of definitions, formulas, and prescribed algorithms. Under these conditions, students engage in memorization and replication and the skills they learn are later quickly forgotten because they were never accompanied by genuine understanding.

Teachers not attempting to flip through every page of their textbook still compromise the learning experience. Their decisions to include, exclude, emphasize, or deemphasize particular content are too often based on their own narrow, shallow, and fragmented understanding of mathematics. Having often mastered only the content of a few grade levels or courses, these teachers are also unable to understand the long-term effects of their decisions. The scope and sequence that students experience reflects the strengths, weaknesses, interests, disinterests, and comfort levels of the teachers more than it does a commitment to a well-defined curriculum map or set of meaningful learning objectives. When critical or prerequisite concepts are underdeveloped and peripheral topics are over- emphasized, it becomes very difficult for students to make smooth transitions from grade to grade or course to course.

Even those teachers who ground their scope, sequence, and pacing decisions upon a commitment to a core set of state and/or national content standards engage students with an anemic pedagogy. Most content standards describe skills that can easily be assessed via the multiple-choice questions of high-stakes standardized testing. Determined to check the teaching of every skill off the list, teachers tend to aim almost exclusively at meeting objectives of the student-will-be-able-to-quickly-and- accurately kind. Mathematics is then reduced to individual students choosing and swiftly executing algorithms, thereby minimizing if not eliminating experiences of investigation, adaptation, discovery, contemplation, collaboration, perseverance and creative problem solving.

American students devote anywhere from 15 seconds to 15 minutes to attempting a single problem. The AB Calculus Advanced Placement exam includes a section composed of three free-response problems to be solved in a maximum time of 45 minutes. On the other hand, for a student to attempt every problem of one mathematics section of the SAT he/she must maintain a pace of approximately 60 to 90 seconds per multiple-choice problem. The 15-minute per problem pace is an experience limited to a select group of advanced students. Most students will resist spending more than a couple of minutes on a single problem. Homework problems that cannot be solved quickly are typically set aside and asked about the next day with phrases such as “I didn’t get number 13.” The teacher typically responds with a full demonstration (and sometimes an explanation) of the proper steps, then asks the student to mimic this example the next time a similar problem is encountered. The student never actually engages in any kind of significant struggle – the kind of struggle that ultimately deepens understanding and sharpens critical thinking skills by requiring students to select, adapt and experiment with various problem-solving techniques.

Many students are now under challenged because they are too quickly excused from wrestling with problems. The teacher hastily concludes that the student is unable to solve a problem when the student shows no signs of knowing the first step to be taken, instead of insisting that the student make some first move, any move, even the wrong move, then another and another and another until he/she has exhausted all options. This tendency to excuse students from the struggle, to save them with a quick explanation, demonstration, and/or prescribed algorithm is not present in the classrooms of top-performing nations. Instead, we see students being called to the front of the classroom to attempt and re-attempt challenging problems, to receive the constructive criticism and praise of their classmates, and to persevere until a solution is found.

This fast-paced learning experience leaves little time for exploration, experimentation, discovery, and argumentation – the processes that were necessary for the development of mathematics across history and are still at the very core of what real mathematicians do. High school geometry is the one course with the greatest potential to engage students in these processes through the challenge of writing mathematical proofs. Yet geometry students are rarely given the time to observe, induce, form, critique, refine, organize, and informally express relationships between numbers and figures. Instead, students are immediately held to a rigid two-column format requiring the very deductive reasoning skills that were neglected and avoided in prior courses. For the students’ sake teachers often limit their consideration to a body of proofs requiring a very limited repertoire of predictable and easily memorized deductive maneuvers.

Thinking creatively and critically, assessing what you don’t know, what you do know, remembering methods/approaches that have worked in the past, anticipating the skills you might use, forming an initial strategy of attack, stepping forward, assessing progress, stepping back then forward again, etc. – these are the experiences that make a study of mathematics most valuable to other disciplines. When these experiences are neglected teachers are then limited to justifying a study of mathematics through immediate real-world applications represented in the form of “word” or “story” problems. Yet the applications are over-simplified, the contexts are far removed from the student’s experience, and the “story” problems rarely tell compelling stories. For many students, word problems such as these simply become annoying syntactic translation exercises that are hardly motivating.

In order to repair mathematics instruction in the United States, teachers must first recognize the primacy of their own content knowledge. If a teacher’s mathematical understanding is fragmented, excessively procedural, narrow, shallow, and decontextualized, then that teacher cannot reasonably expect the understanding of his/her students to become any different. In order for a teacher to navigate his/her way through oversized textbooks, identify and emphasize essential content, design lessons around meaningful learning objectives, engage students in critical thinking, appropriately represent and value the struggles of problem solving, and teach mathematical reasoning he/ she must know the content well – not just the content of a single grade level or course but rather the entire K-12 curriculum.

Should the teaching practices identified by recent research be corrected by teachers recommitted to increasing their content knowledge and refining their pedagogy, the United States might rise in the rankings but the mathematical experience of students would still remain incomplete. Why? Because our obsession with international competitiveness has caused us to completely neglect those immeasurable characteristics of mathematics that make mathematics most lovable. Coherence and contextualization must be gained through the integration of the historical narrative – its master works and colorful personalities. Students must have their attention drawn to universals encountered in mathematics, truths indicative of something eternal and greater than the mathematics itself. Time must be devoted to contemplation of the sheer beauty of mathematics. Mathematics must be recognized as a language inexplicably able to describe natural phenomena, enabling us to better understand God’s creation and praise Him for His majesty.

Let us repair these ruins.

Principles and Elements of Effective Professional Development

Posted on by Society for Classical Learning

The Biblical concepts of sanctification and community will set the foundation for a professional development plan that both encourages all teachers and challenges teachers to higher levels of professionalism. An overview will be given of a plan composed of: peer and self observation, collaborative lesson design (a.k.a. Lesson Study), a four year cycle of readings and discussions, critical friends tuning protocols, smaller team/cluster meetings, weekly email updates, monthly collegiums, annual traditions, and summer conferences. A corresponding packet of sample documents will be provided to participants.

Andrew Elizalde

Andrew Elizalde earned his B.A. at Depauw University where he graduated Phi Beta Kappa and Magna Cum Laude, earned a math major, physics minor, and religious studies minor, and received the H.E.H. Greenleaf award as the most outstanding 2000-2001 graduate of the school’s mathematics program. He later obtained a teaching credential from California State University Long Beach with a professional- clear qualification for his coursework regarding exceptional children and technology integration. His teaching experience includes work in both public and private schools in subjects ranging from 5th grade mathematics to advanced calculus and physics. Andrew now serves as the Dean of Academics, Mathematics Department Chair and Lower School Principal at Veritas School, a K-12 classical Christian school in Richmond, VA. His work at Veritas has most notably included a comprehensive reform of the schools’ K-12 mathematics program, the design and implementation of annual in-house professional development, and the advancement of a student support services plan. Additionally, Andrew offers consulting services to classical Christian schools aiming to refine pedagogy, mathematics curriculum, and professional development strategies. He has been a keynote speaker at the ICS Math and Science Lyceum and most recently, Trinity Classical Academy’s annual conference in Southern California. Andrew and his wife have three daughters and are members of All Saints Reformed Presbyterian Church.