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Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined. Bright Students: Top universities want you. While many have high fees: many will lower them, many will provide funds, many have more scholarships than students. Postage is cheap. Apply and ask how much help is available. Caution: some programs are rewarding. Others lead nowhere. After acceptance, it may be easy or not to switch. For students of reason in society, science and technology: Pattern Based Reason describes origins, benefits and limits of rule- and pattern-based thought and actions. Not all is certain. We may strive for objectivity, but not reach it. Postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theories and practices. Site's Best LessonsFor Logic
These online chapters may amuse while leading to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics. For Arithmetic
Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style. For Algebra
What is
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
correct? |
www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Chapter 12 Islands-and-Divisions-of-Knowledge Next: [Chapter 13 Geometric-Thinking-Euclidean-Model-For-Reason.] Previous: [Chapter 11 Accidental Patterns.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14][15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] Chapter 12, Islands and Divisions of KnowledgeRecall the difference between one- and two-way implication rules:
The examples in the chapter Chains of Reason involved one-way implication rules. They showed that one-way implication rules can sometimes be put together to get further implication rules. You may remember we had one implication rule about Charles that was not used to get any conclusion. Two analogies (Isolated Islands or Ignorable Rooms)One and two-way implications can also be joined. The ways in which this can be done are described below by analogies with one- and two-way streets, and one- and two-way doors. These analogies indirectly describe how rule-based knowledge is put together. In particular, rule-based knowledge is divided into separate segments. Each segment cannot be reached from another by chains of reason. The two analogies describing this situation further are presented next. Islands Without Roads BetweenImplications are like streets or roads. They may be traveled one-way or both ways. Streets (or implications) may lead nowhere. Others may lead to interesting and sometimes unexpected places. Each road may touch several others. Each of these others may touch several more. But by foot or car, from one road, there is no guarantee that all roads can be reached. Moreover, when some one-way roads are present, poor planning may imply no return route for every possible starting point. Maps make the exploration of any road system easy. All we have to do is read the map. Without a map, we have to explore the neighborhood in which we live, and hope we can find a path back. One-way streets are a danger here, unless another path back is available. Without a good map, we cannot say in advance, when we explore the streets, if we will get to an interesting or boring destination. To find out what is interesting, our only choice is to explore or to ask whether any one has made a map. We would like to learn from the experience of others, perhaps. By road, not all destinations are accessible or reachable. We may for example have roads on several islands with no boats, ferries, planes, bridges or ships to take us between them. Without boats, ferries, planes, bridges, or a very low-tide, we have no route or connection between one island and the next. Without these extra routes, the roads (or implications) of one island are not linked to the roads of another. The streets on even a single island need not all be connected to each other. For example, imagine on one island that a mischievous or artless road planner has provided one-way roads all leading from one end of the island to the other. On such a road system, a return to the starting point is not possible. We can imagine another island in which the planner, mischievous or not, has placed a mixture of one- and two-way roads. From some starting points you can leave but not return. From some parts or destinations, you cannot leave. Between other starting points and destinations, you can go back and forth. And after going back and forth several times, you may forget which place was the destination or the starting point. All the situations just described with one- and two-way streets can happen similarly in logic with one- and two-way implication rules. In other words, knowledge is linked by one- and two-way implication roads, spread over several islands. The map of this area is not complete. As we explore and forget, roads and routes new to us or our neighbors are uncovered or rediscovered. Rooms Without Doors BetweenImplication rules are also like doors or gates between sections of a building or estate. (Implication rules like doors join the rooms of a large palace, castle, house or prison. ) Some allow two-way passage. Others permit only one-way passage. All this can be a deliberate design or it could be due to a poor design. When we restrict our paths to two-way doors, we can always retrace our steps exactly and get back to where we started. But one-way doors are different. To get back after going through a one-way door, we need to find another route back through some other door or doors. Otherwise, we are shut out of our starting room. That is, we suppose a one-way door can only be opened from one side, and that after use it snaps shut. When we go through a one-way door, we can get back to our initial side of the door only if there is a route back. But by passing through one-way doors, we may find ourselves locked out of the initial room we were in. We may further find ourselves locked in another room or section of the building. Ignored RoomsWhenever the building we are exploring has sections closed off or unreachable, we can ignore all maps of those sections. Making a map of the unreachable sections is not possible, except by guessing. Guessing is suggestive, yet not reliable. Selby A, Volume 1A, Pattern Based Reason, 1996. www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Chapter 12 Islands-and-Divisions-of-Knowledge Next: [Chapter 13 Geometric-Thinking-Euclidean-Model-For-Reason.] Previous: [Chapter 11 Accidental Patterns.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14][15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] |
Road Safety Messages for All: When walking on a road, when is it safer to be on the side allowing one to see oncoming traffic? Site Reviews1996 - Magellan, the McKinley Internet Directory: Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it. 2000 - Waterboro Public Library, home schooling section:
CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as
how (and why) to learn mathematics in school; pattern-based reason;
finding a number; solving linear equations; painless theorem proving;
algebra and beyond; and complex numbers, trigonometry, and vectors. Also
section on helping your child learn ... . Lots more!
2001 - Math Forum News Letter 14,
... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot- and
cross-products, the cosine law,a converse to the Pythagorean Theorem
2002 - NSDL Scout Report for Mathematics, Engineering, Technology -- Volume 1, Number 8
Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and how-tos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.
2005 - The NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4, Number 4
... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...
For Geometry
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
to scale. For Calculus
Why study slopes - this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals. |