Mathematics and Logic - Skill and Concept Development

Mathematics with Take-Home Value

Essay, Draft form.

Before a child goes to school, many mathematics, logic and language skills will appear in home life. Which ones have the most take-home value may vary. Different communities may have different needs. Some may serve practical ends. Many skills are required in buying and selling goods, property and services, in making informed decision in the presence or absence of certainty. Practical applications done and recorded in an observable and verifiable or correctable manner set an empirical standard for rigour, one in which the domino effects of wishful thinking and mistakes have immediate consequences.

We need to remember that the time in school of a child or teenager may be limited because of family circumstances or because of a voluntary termination of schooling. Thus Mathematics skill development should focus in the first instance on what TCPITs, the common person in the street, may need sooner or later in daily or adult life. There-in lies a place for a broad development of mathematics and logic skill with take-home value. That development should begin before the preparation for college studies, but may overlap because the latter preparation in part may make skills with take-home easier to learn and teach.

In daily life, many skills are mastered and used without a full comprehension of how and why they work. In preparing a meal, very few if any of us full understand the digestive system. Machines and mechanisms may be also be employed without comprehension of how they work and why. That being said, in counting, figuring and measuring, the ability to do is required while comprehension is optional. In the case of methods with take home value, the first task is to provide skill with methods without overwhelming students with explanations of why, but explanations included to the extent that they aid skill development, but do not overwhelm it. Skill mastery is the first aim of instruction in methods with take-home value.

Where primary instruction in methods with clear take-home value overlaps secondary instruction preparing students for college and also prepares students for further skills that have take-home value, Primary instruction has skills with take home value to development explanations of how and why rules and patterns give results is must to the point that it aids skill development without overwhelming a student. Beyond that, where secondary instruction preparation for college studies in business, engineering, science, technology and mathematics education should gradually emphasize comprehension of both how and why, with the explanation that skill without comprehension becomes limited. That being said, the theory or explanation of how and why should be lean. Where topics do not have immediate take-home value, reasons for them should be given. See site account of secondary ends and values - those to emphasize in college oriented skill development.

A. Time and Date Skills

Modern family life may be arranged in accordance with the time of day, day of the week, the month, the season and the year. Every one has a birthday. Year round, daily life is governed by seasons and the day of the week, what time to rise or eat, what time to sleep, when to work and study and pray. School and work life are run or organized around time or schedules. Events occur before, after or the same time. Civic and religious calendars provide schedules, celebrations and holidays. The teen or adult may have study and work schedule as well.

B. Money Skills

People buy and sell goods and services. People work for a living, There are budgets to balance in private life and in business. Money is counted, added, compared, subtracted, multiplied and divided. Life in families and communities often involves money matters. Talking about ends, values and methods for handling money at home, and in buying goods and services, or balancing a personal or business budget would provide a context and motivation for care and diligence in arithmetic with unsigned and even signed numbers - the use of the latter is optional. Students may shown how calculate net worth by adding subtotals. See corresponding remark below.

In the case of money matters, students may master the forward and backward use of compound interest/growth formulas and geometric summation formulas by rote initially, simply because of the possible take-home value. For example, numerical examples may be employed to empirically suggest or confirm the geometric sum formula. To ease or avoid raising the level of algebraic complexity, formulas may given and illustrated instead of being derived independently or from each other. In the first instance, the ability to apply methods with take-home value in a repeatable and reproducible manner is still more important than know-why.

C. Spatial Skills

Child may learn to recognize the spatial positions of before, after, behind, ahead, above, below and besides. That learning may come from experience or from others at home and in class. The tutor or teacher may check or develop this knowledge through simple questions and activities.

The spatial sense of up and down, or above and below, may be easier to learn and remember than the sense of left and right.

D. Geometry, Navigation and Construction Skills with maps and plans

Maps and plans drawn to scale (or not) are everywhere. In going to school and in traveling children may see maps and plans. Larger schools, colleges and workplaces may use building plans or maps to provide directions.

Geometry in part consists of direct measurements with units of length, on rulers and tape measures, with angle measures in terms of degrees and fractions of a circle or revolution. Students may identify a right angle with a quarter of circle before thinking about quarter revolutions. Scaling of Direct Measurement of lengths and angles on maps and plans drawn to scale then gives another method, not direct, for finding and using lengths and angles not measured directly. Triangle construction algorithms ASA, SAS, SSS and even AA(A) will be included among methods for drawing triangles, rectangles and the circles to scale or full size.

For geometry the foregoing students may learn on paper to evaluate a formula for an area or volume given the necessary lengths and measures. They should also be able determine those lengths or measures from hands-on experience with the actual figure or scale drawings of it. Instruction should point out applications robustly and fully.

E. Counting, Measuring and Figuring Skill

Besides counts and measures of time, dollars and length, master measures of area and volume, and of mass and weight. In buying and selling goods and services, and in making things - that includes cooking and construction - people use and combine measures alone and in proportions. Examples include speed and the cost of a unit (a prequel to the discussion of per unit rates) of a good or service.

1. Express Measure & Counts in terms of given and alternate units of each, respectively.

2. Add, Compare and Subtract Counts and Measures - when possible.

3. Multiply: Form the product of both with a number.

4. Divide Measures & Counts by another measure or number.

5. Each rate has a reciprocal. For example, the rate of 5 apples per 10 pennies is equivalent to the reciprocal rate of 10 pennies per 5 applies. The transformation of one rate into reciprocal may be taught along side the use of rates in arithmetic. Mastery here may be verbal. Skill here can be extended with the arithmetic with multiples of units and fractions with such mutliples in numerators and denominators.

6. Example of the chain rule for rates is given by observation that 5 apples for each 10 oranges, and 10 oranges per 20 pears implies 5 applies per 20 pears as well. With the mastery of arithmetic with fractions, the chain rules can associated with a product of fractions with units.

Exposure to measurement matters in the home and in shopping will depend on the family and community life of students. Due to the variation, skill development in school will have provide the missing experience and a context for it. But all has take-home value.

F. Chance and Risk Skills

In decision making, not all is certain. Risks are present even for people who avoid games. Learning about chance and probability may help avoid situations or decision where the risk are high, or help in making decisions that lower risk and make the chances of success greater. Again, due to the variation, skill development will have provide the missing experience and a context for it.

G. Logic and Decision Making Skills

Logic mastery in mathematics or apart - say in a reading and writing course - may lead to greater care or precision in reading and writing, and so avoid or lessen difficulties in studies and in work. Precision in reading and writing may follow here from study of logic and from an awareness of the domino effect of errors.

The leading chapters of Volume 2, Three Skills for Algebra, develop deductive reason (logic mastery) in a math-free way. Altogether, those chapters hint at the partial Euclidean organization and codification of rule and pattern based arts and disciplines.

Awareness of the difference between say A if B and saying A if and only if B (or equivalent expressions) will sharpen reading and writing. And seeing how to chain implication rules together will help reasoning in general. These two elements of logic and further elements may be introduced when students are ready for them - the age level for that may depend on the student. The net result should fewer difficulties in work and study, and better results in general.

Self-Defense

People in health care and instruction may sense an obligation to help others. But in buying and selling goods, properties and services, and in working for a living, decision may require money skills and logic or language skills to negotiate, to recognize the good and bad aspects of deals and contracts, and to recognize agreements that should not be made.

H. Proof

The aim of instruction in the above areas and in the parts of arithmetic they require is to develop observable and verifiable skills and know-how. As part of that, we will show learners how to do and record numerical and geometric steps in learn but sufficient show work formats that allow steps and results, from start to end, to seen and confirmed or corrected. Emphasizing that care and patience, and precision too, are needed to avoid the domino effect of errors and approximations in short and long chains of reason will be emphasized. The ability to figure well, and to read and write with precision, is an observable sign of intelligence of the practical kind for work and studies in general.

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