The official textbook for the course adopts a fairly traditional approach to thinking about mathematics. Basic ideas are formally introduced in an order that is established mainly by logic. Problems and ideas are introduced in advance of when they would normally appear naturally, and so these problems feel out of place when first introduced — they feel arbitrary and often confusing. In addition, many of the introductory ideas will be ideas that you have seen before. But this does not mean that you understand the ideas at the level necessary to move forward. Unfortunately, familiarity may lead you to adopt an overly casual approach to thinking about this material.
Researchers have studied how people learn. The National Academies of Science published a book, How People Learn, (National Academies Press, 2000) that has motivated most of the content of this page. The full text is publically available at the National Academies Press website.
One of the most important conclusions from this research is a recognition of the differences between novices and experts. Over a wide range of disciplines and skills, a common collection of attributes provide a way to classify whether an individual is a novice or has attained the status of expert .
A second important aspect of the research about how people learn is the role of metacognition. Metacognition is the process of thinking and reflecting about the way we think and learn. When experts were asked to verbalize their thinking as they worked through a problem or learned a new concept, they showed that they monitored their own understanding, noting when things did or did not relate to what they anticipated, when they needed to seek for additional information, and how they could develop new organization structures for their thinking (analogies). You can improve your personal model of learning by incorporating metacognition as an integral aspect of your learning process.
The most important component of this is ownership of your education. That is, you take full responsibility for learning. While I, as your teacher, take seriously the charge to provide meaningful instruction, activities, and assessment, it is not as if I can just pour knowledge into your head and make it organize itself properly. Furthermore, you are not the only person in the class, and it is impossible for an instructor to remain 100% aware of every student's progress and understanding. If you finish an assignment and do not yet feel you understand, then personal ownership of your learning should lead you to seek additional opportunities to learn.
Metacognitive learning involves establishing learning goals and then monitoring your progress in achieving those goals. You should practice explaining to yourself what you are trying to accomplish, how you do it, and anticipate what kind of result you will get. You should be fully self-aware of when you fail to comprehend something and learn to identify the limits of your current understanding. You need to plan ahead for adequate time to master the necessary objectives and for adequate training of your memory and recall for the knowledge to become permanently engrained rather than temporarily crammed.
Your ability to incorporate new knowledge depends on how your existing foundation of knowledge has been constructed. We all make sense of new information according to the context of what we already know. The Russian psychologist Vygotsky proposed the idea of a zone of proximal development (ZPD). This zone describes the collection of new knowledge and skills that an individual is prepared to learn with additional support. Your ZPD will likely differ from other students. Furthermore, as your knowledge increases, this zone will expand to new material.
Attempting to learn something outside of your ZPD will lead to frustration because you are not adequately prepared to add the knowledge or skill to your foundation of knowledge. Having taken the proper prerequisite courses is not enough to guarantee that your ZPD is properly prepared. Your foundation might include the basic knowledge but it may not been organized effectively. Such a foundation needs to be reconstructed in a more effective structure. Alternatively, your foundation might be built with misconceptions and faulty strategies. This time, the misconceptions need to be replaced with the correct foundation material and skills. In both cases, metacognition will play a significant role in determining a strategy for remediation.
Let us now take a look at the textbook, Calculus I with Integrated Precalculus by Laura Taalman (2nd edition, W. H. Freeman, 2014). Math 231 deals with the first half of the material, including chapters 0 to 4. The design of this textbook breaks each section into two basic components: (1) conceptual material and (2) examples and explorations. The problems are broken down into the same two categories, with (1) concepts and (2) skills along with applications and proofs. You should use this structure in the book to guide your learning and self-assessment of understanding. Use the suggested exercises as guides for your personal educational goals for the course.
Chapter 0 is a foundation chapter. In principle, it contains all of the basic prerequisite knowledge and skill necessary to be successful in the remainder of the calculus course. Mathematically, the organization of the chapter is appropriate. Content is organized by topic, making this chapter a useful reference. But as your instructor, I'm not convinced that it is properly organized for learning. So we will be addressing the pre-calculus prerequisite information in a modified format.
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