www.whyslopes.com || Fit Browser Window

Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

30 pages en Francais || Parents - Help Your Child or Teen Learn
Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

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.


Are you a careful reader, writer and thinker? Five logic chapters lead to greater precision and comprehension in reading and writing at home, in school, at work and in mathematics.
- 1 versus 2-way implication rules - A different starting point - Writing or introducting the 1-way implication rule IF B THEN A as A IF B may emphasize the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
- Deductive Chains of Reason - See which implications can and cannot be used together to arrive at more implications or conclusions,
- Mathematical Induction - a light romantic view that becomes serious.
- Responsibility Arguments - his, hers or no one's
- Islands and Divisions of Knowledge - a model for many arts and disciplines including mathematics course design: Different entry points may make learning and teaching easier. Are you ready for them?

Early High School Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6.
- Decimals for Tutors - lean how to explain or justify operations. Long division of polynomials is easier for student who master long division with decimals.
- Primes Factors - Efficient fraction skills and later studies of polynomials depend on this.
- Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for addition, comparison, subtraction, multiplication and division of fractions.
- Arithmetic with units - Skills of value in daily life and in the further study of rates, proportionality constants and computations in science & technology.

Early High School Algebra

What is a Variable? - this entertaining oral & geometric view may be before and besides more formal definitions - is the view mathematically correct?
- Formula Evaluation - Seeing and showing how to do and record steps or intermediate results of multistep methods allows the steps or results to be seen and checked as done or later; and will improve both marks and skill. The format here allows the domino effects of care and the domino effects of mistakes to be seen. It also emphasizes a proper use of the equal sign.
- Solve Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to present do and record steps in a way that demonstrate skill; learn how to check answers, set the stage for solving word problems by by learning how to solve systems of equations in essentially one unknown, set the stage for solving triangular and general systems of equations algebraically.
- Function notation for Computation Rules - another way of looking at formulas. Does a computation rule, and any rule equivalent to it, define a function?
- Axioms [some] as equivalent Computation Rule view - another way for understanding and explaining axioms.
- Using Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards. Talking about it should lead everyone to expect a backward use alone or plural, after mastery of forward use. Proportionality relations may be use backward first to find a proportionality constant before being used forwards and backwards to solve a problem.

Early High School Geometry

Maps + Plans Use - Measurement use maps, plans and diagrams drawn to scale.
-
- Coordinates - Use them not only for locating points but also for rotating and translating in the plane.
- What is Similarity - another view of using maps, plans and diagrams drawn to scale in the plane and space. Many human-made objects are similar by design.
- 7 Complex Numbers Appetizer. What is or where is the square root of -1. With rectangular and polar coordinates, see how to add, multiply and reflect points or arrows in the plane. The visual or geometric approach here known in various forms since the 1840s, demystifies the square root of -1 and the associated concept of "imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.
- Geometric Notions with Ruler & Compass Constructions :
1 Initial Concepts & Terms
2 Angle, Vertex & Side Correspondence in Triangles
3 Triangle Isometry/Congruence
4 Side Side Side Method
5 Side Angle Side Method
6 Angle Bisection
7 Angle Side Angle Method
8 Isoceles Triangles
9 Line Segment Bisection
10 From point to line, Drop Perpendicular
11 How Side Side Side Fails
12 How Side Angle Side Fails
13 How Angle Side Angle Fails

Return to Page Top


www.whyslopes.com >> Mathematics Skill Development Framework >> Multiple Ways to Improve Mathematics Skill Development Next: [Ends Values Methods For Skill Development - Framework Prequel.] Previous: [ Phase 5. Calculus Light to Rigourous for 1 to 2 years.]   [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13][14] [15] [16] [17]

Multiple Ways to Improve Mathematics Instruction

    From counting 1-2-3 to proofs, problem solving and calculus, skill in mathematics may be defined by showing students and teachers how to do and record work in steps that may be seen as done or later for confirmation or correction.

    In learning to figure well, students will see the domino effects of mistakes, and thus become more careful and cautious in applying and learning skills.

    In problem solving, thinking out of the box is fine, but that box should be as large as possible. Thinking out should not be needed for routine problems.

My online endeavours to change mathematics instruction stem from observation of gaps in the exposition of algebra. All ways but the last two represent practices to may change instruction. The last two represent ends and values, or theory. So this communicatin put practice first, theory second. Each way or possibility below reflects ideas to vulgarize instruction and to set a higher lower bound for skill development.

In my offline endeavours to talk about the need for small changes, my colleagues had little patience. In their student days, their professors had set the model. That and years or decades of practice in the classroom implied my efforts were not required. Likewise, self-taught by trial and error in Nordic skiing a long time ago, I had the same perpsective until a short course on how to be a Nordic ski instructor. In it, my eyes were open.

The mathematics education I had seen in school appeared to have steps too large and missing in algebra skill provision. There were inconsistencies too between the content of secondary and college level modern mathematics skill and concept selection, and the needs [a] of life in the street for common decimal skills and concepts to be explicitly mentioned and sanctioned; and the needs [b] of arithmetic and algebra with denominate numbers - that is multiples of units of measure - to be explicitly mentioned and sanctioned for life in the street and for use in science and engineering subjects. The need for explicit mention and sanction was doubled in courses in which emphasized rigour directly or in asides. And in retrospect, the axioms for algebra and geometry were not fully, that is coherently, tied together in my secondary and college level, modern mathematics education.

The past is not a perfect model for the future. Departures ranging from better or more efficient practices for introducing key topics to including more axioms and assumptions to make learning and teaching easier are possible.

1. Quick Prime Factorization and Recognition. The ability to recognizing primes and obtaining prime factorization of of whole numbers less than 169, even 100, is sufficient for most ends and purposes purposes in high school and college courses. To that end we may use the following rule in a recursive manner.

If a whole number N less than 169 = 132 is not divisible by 2, 3, 5, 7 nor 11 [the primes smaller than 13] then the number is prime. Otherwise, it - the number N - is a product of 2, 3, 5, 7 or 11 with another factor N' less than 169.

Knowledge of times tables, divisibility rules or a calculator - one displaying results to two plus decimals, may be used to recognize multiples of 2, 3, 5, 7 and 11. Two decimals are sufficient because 0.09 is less than each of the fractions one half to one eleventh.

The above rule is a special case of the following.

If a whole number N less than the square of a given prime is not divisible by all the the primes smaller than given one then the number is prime. Otherwise, it - the number N - is a product of one smaller prime and another factor N' less than the square of the given prime.

It too can be used in an recursive manner. Inspection of the 12-times table implies 2, 3, 5, 7, 11 and 13 if we say that a whole number is prime if it is not the product of two smaller whole numbers. Equivalent definitions of prime numbers are available. This one appears to be the most convenient for vulgarization.

    In prime factorization, the grouping of like primes represent in practice an tacit extension of commutative and associative laws. That tacit extension is present in the next topic as way.

2. Raising Terms to Multiply and Divide Fractions. Raising terms to add, compare and subtract fractions is a standard method to introduce or show how to do these operations. Raising terms can also be used to say how to multiply and divide fractions in a vulgar. In that, just as three quarters is three times a quarter, the fraction $\frac ab = a \times \frac1b$. With small whole numbers n, a and b, the identity $n \times \frac ab = n \times [a \times \frac1b] =\frac{n \times a}{b} $ may be implied by example

For multiplication, observe an $n$-th of the multiple $n \times a$ of $n$ is $a$. Now each fraction by raising terms \[\frac ab = \frac {n \times a}{n \times b} = n \times a \times \frac 1{n \times b}\] equals a multiple of $n$. One n-th is \[ \frac 1n \times \frac ab = a \times \frac 1{n \times b} = \frac a {n \times b} \] So m times an n-th of that fraction would be m times as much \[ \frac mn \times \frac ab = m \times a \times \frac 1{n \times b} = \frac {m \times a} {n \times b} \]

For division, the number of times D units goes into N units is the fraction $\frac ND$. Here we think of units as a kind of object to count and to divide. Clearly, \[\frac ND\times D \mbox{ units } = \frac {N \times D} D \mbox{ units } = N \mbox{ units } \] by lowering terms.

The number of times a fraction $\frac MN$ goes into a fraction $\frac PQ$ may be demonstrated by placing both over a common denominator $N \times Q$. With one unit taken to be $\frac1{NQ}$, that route gives

\begin{eqnarray*} \frac MN = \frac {MQ}{NQ} = MQ \mbox{ units} \\ \frac PQ = \frac {NP}{NQ} = NQ \mbox{ units} \end{eqnarray*}

Therefore the number of times a fraction $\frac MN$ goes into a fraction $\frac PQ$ is \[\frac MN \div \frac PQ = \frac{MQ}{NQ} = \frac MN \times \frac QP \]

3. Complex Numbers Before Trigonometry - a vulgarization.

We may introduce complex numbers as points in the plane. That identification allows complex number a + i b to be identified by rectangular coordinates [x,y] and polar coordinates $(r, \theta)$. The geometric correspondence between rectangular and polar coordinates may be tacitly assumed here. For the sake of vulgarization, the moduli and arguments terminology for complex numbers can be introduced later. With these three representation of complex numbers, Then sums may be defined using rectangular coordinates \[ [x_1,y_1] + [x_2,y_2]= [x_1+x_2, y_1+y_2]\] while products may be defined using polar coordinates \[ (r_1, \theta_1)\cdot (r_2, \theta_2) = (r_1r_2, \theta_1+ \theta_2) \] But for the distributive law, all the field properties of complex numbers follow the field properties of real numbers. The distributive law itself follow from two observations.
  1. Scalar multiplication by $a$ distributes over addition. Here multiplication by $(r_1, 0)$ is equivalent to a scalar multiplication.

  2. Rotation distributes over the calcultion of midpoints of line segments. Why can be explained as follows.

    In general, two points $[x_1,y_1]$ and $[x_2,y_2]$ in the plane with the origin determine a triangle with vertices at $2[x_1,y_1]$, $2[x_2,y_2]$ and the origin $[0,0]$. The midpoint of the side opposite the origin is given by $[x_1+x_2, y_1+y_2] = [x_1,y_1] + [x_2,y_2] $. Rotating the triangle about the origin rotates the midpoint of the side opposite the origin the original triangle into the midpoint of the image of that side under rotation. Therefore mid point calculation for the triangle and its image commutes with rotation. From which may say rotation distributes over addition.

The equality $(r_1, \theta_1) =(r_1, 0)\cdot (1, \theta_1)$ together with the associative law implies pre-multiplication by $(r_1, \theta_1)$ is equivalent to a rotation followed by a scalar multiplication. Since both transformations are distributive, so is their composite. That implies the left distributive law. The right distributive law follows from commutativity.

In a further introduction and study of trigonometry and vectors in the plane, the equality of polar- and rectangular-coordiante ways to calculate products easily implies trigonometric identities and also formulas for dot- and cross-product of position vectors in the plane. Each vector locates a complex number. The product of one with the complex conjugate of the other can be calculated in two different ways, equality of which implies the formulas.

Algebra Troubles. Arithmetic and algebraic expressions and formulas are often too complicated to read aloud, term by term. Operations in algebra, geometry and calculus may be done in silence. The oral dimension to mathematics has been missing. The algebraic shorthand role of writing and reasoning in often just appears over many years with a sink or swim option. People may say algebra skill is a natural talent which students have or lack, which may surface eventually. And in those that succeed, it does. But the oral dimension of mathematics can be expanded and refined. The next steps show how.

5. Using Names and Descriptive Phrases. Many arithmetic and algebraic expressions, axioms included, are best seen and understood in silence. The compound formula and quadratic formula are hard to read aloud. The description of the associate calculations with words would take a thousand, and not necessarily be as clear. Formulas like pictures are sometimes worth a thousand words. Yet in speaking about a rectangle area calculation, the sphere volume formulas, the quadratic formula or even the painting the Mona Lisa, descriptive phrases and names may bring the formulas or painting into a our minds. Phrases and names allows to handle or at least talk about objects. The phrases and names obviate the need to see the formulas and pictures. That enlarges the role of words in mathematics.

6. Describing Calculations and Arithmetic Properties. As mathematicians, we are in the habit of describing calculations and identities with algebraic expressions. But in the vulgarization and teaching of mathematics, there calculations and identities easy to understand and describe with words. For a first example, perimeter calculation for polygons described algebraically may introduce letters alone or with subscripts to denote the length of sides as a part of a sum giving a perimeter. But the far simpler way to describe the calculation consists of the instruction or slogan: Add the lengths of the side. The latter is brief and for many students, clearer than any formula with collection of letters alone or with subscripts.

For a second example, book-keepers may sum assets and debts via subtotals. The subtotals may be themselves be given subsubtotals. The grouping and subgrouping of terms in sum does not affect the result. That useful practice is easily described with words and illustrated by examples, but a generalized associative law would be need to describe the practice or the underlying identity algebra. We would not forbid that practice because the book- keeper has not taken an advanced courses in mathematics. In primary school, students may be shown the related practice of grouping and subgrouping for sake of counting by adding subcounts.

For a third example, multiplicative instead of additive, prime factorization practices met before algebra group like primes in powers and in ascending or descending order. As with addition, the grouping and subgrouping of factors in product to express it a product of subproducts, perhaps in a recursive manner, does not affect the result. The practice is easier to describe with words than it is algebraic. The two practices, verbally described, may employed in the study of polynomial sums and products. In my modern mathematics school days, I remember appreciating but not fully following a teacher's effort to quickly explain with the aid of axioms for real numbers, what justified addition and multiplication operations though reference to axioms for real numbers. The two adding and multipling by grouping practices would have made the exposition clearer.

Instruction aiming to provide students with an operational command of skill and concepts is not restricted to deriving all from a minimal set of axioms, algebraically stated. For vulgarization, instruction may provide a growing collection of consistent rules and patterns while emphasizing for intellectual value how some imply the rest. There-in lies a relaxed deductive framework.

7. What is a Variable. Apart from the use of letters and symbols to denote numbers, amounts and quantities, we may describe the latter. For example, we may speak of the height of a bird above the ground without using a letter to denote it. This height may be described as constant, known, unknown, measurable or changing and variable. This verbal element of mathematics is apart from any representation of the height by symbols and apart the more advanced discussion of functions. In the vulgarization of mathematics, the height of a bird as it flies is a variable and on landing the height becomes constant. There is a pre-algebraic concepts of what is a variable or what is constant. Now if we use a symbol like H for this height and its value, we say H in lieu of the height is constant, variable, unknown and so on. The perspective here is clearer than the nonsense in some dictionaries which say a letter in mathematics is a variable. The perspective here can be precursor to the modern mathematics or logic views of variables as placeholder or variables a functions.

8. Computation Rules and Algebraic Explression. Letters in areas, volume, compound growth and quadratic formulas typicallly have clear meanings. Yet the computational significance of letters in expressions like x, $y^3a^2$, $5zb^3$, $x^4$ may be invisible to students. We may counter that explicitly associating computation rules. For the foregoing examples, that might give \begin{eqnarray*} f(x) = x \qquad &g(a,y) = y^3a^2 \qquad &h(z,b)=5zb^3 \\ \qquad & p(x) = x^4 \qquad &A = P(1+r)^n \end{eqnarray*} Exercises in the evaluation of computation rules by hand, with the aid of calculators or with the aid small computer programs would assign a role of placeholders to the letters or variables serving as arguments in these computation rules.

9. Computation Rules and Axioms. In geometry, the distributive law $a(b+c)=ab+ac$ may represent or follow from the equality of two different ways to find the area of an rectangle with height a and length b+c. At the same time, the distributive law may be seen as the statement that two different computation rules $f(a,b,c) = a(b+c)$ and $g(a,b,c) = ab +ac$ will give the same result for all real numbers a, b and c. The two computation rules f and g are represent different calculations, but they define the same function. In general, a computation rule determines a single function, but a single function or its values may be given by several different computation rules or methods. The commutative and associative properties of addition and multiplication cast in the same framework to help students understand axioms or properties of real numbers.

10. Solving Linear Equations with Stick Diagrams. Students may initially be more comfortable with a letter x denoting the length of a line segment instead of unknown numbers if the above ideas on computation rules have not been met. My work includes a three column stick diagram method as a temporary measure to introduce students to solving linear equation in one unknown. The first column is used for physical addition, subtraction, multiplication and division operation on the stick or line segment, the middle column is use to describe the operation, and the third column is used to present the algebraic steps. The physical operations on the sticks may develop and test fraction skills and sense. The stick approach may quick lead students to understand the solution of linear equations in one unknown with fractional coefficients first with and then without stick diagrams. The without is the objective. Students who can solve linear equations without will only benefit from the reinforcement of fraction skills in the approach. The method when tried in class was usually effective.

11. Solving Linear Equations - More Steps. Once students can solve linear equations in one unknown, their algebraic skills may be easily expanded by showing how to solve triangular systems and systems in essentially one unknown. Many first and second year high school word problem are more easily written as a system of equation, easily recognizable a system in essentially one unknown. So algebra instead of mental gymmastics may be used to obtain an single equation in one unknown. Whence the solution of word problems becomes simpler. Finally, the substitution methods for solving systems of equations in essentially one unknown illustrates and a sets the stage for Gaussion elemination in more general systems by substitution. Thus skill development may take smaller step to make the general more accessible.

12. A Unifying Oral Theme for Algebra, Logic and Calculus. To introduce the algebraic or literal solution of equations, my online work emphasizes the common numerical patterns in solving many very, very similar numerical versions of a problem before rewriting and capturing the underlying pattern with letters instead of numbers. Examples of this process and its great but deliberate redundancy, appears online in my introduction of literal solutions of the linear equation $ax+b =c$ for and of the compound interest formula $A=P(1+i)^n$ for P, i or n. The examples is to introduce students to the power of algebra in solving may problems of the same form at once by developing a formula. In UK terminology, in the compound interest formula $A=P(1+i)^n$ , the final amount A is the subject. But given A and two of the three quantities P, i and n, the literal derivation of a formula for the missing is called changing the subject. In my work, I call that a backward use of a formula. Talking about the forward and backward use of formulas, rules and patterns, each time the backward use arises, will lead students to expect it and not see each backward use in isolation.

In proportionality relations, the constant of proportionality is usually found first from given data. That represents a numerical backward use of a proportionality equation. Next the proportionality equation may be used forwards or backwards to obtain the numerical value of other quantities. By changing the subject, proportionality relations may imply each other. In the analysis of similar geometric shapes, simultaneous proportionality relations appear between the lengths, areas and volumes of corresponding parts, with proportionality constants in each be related. The forward and backward use of these simultaneous relations may go in many in many directions.

In logic, the backward use of an implication rule If A then B gives If NOT B then NOT A. In chemistry, physics and calculus most of the equations and rules will be employed backwards and forwards. In the study of functions, the backward use of the formula y = f(x) in solving for x gives the inverse. In calculus, the backward use or twist of differentiation rules gives integration rules. The forward and backward use of formulas, proportionality relations and more generally, rules and patterns of all types, may be serve as a unifying phrase and theme in mathematics and science, and in doing help extend the oral dimension of mathematics skill development.

13. Maps, Plans, Diagrams and Geometry.

Much geometry can be done with maps, plans and diagrams in place of direct measurement. The use and drawing of maps, plans and diagram done to scale represents a common application of and context for similarity principles and practices present Applications may range from reading and making plans for clothes, floors and gardens to location of points and navigation. Applications may involve measurement of angles, lengths and areas with or without scale factors. On maps and plan may give students an operational command of geometrical practices in the use of scaled drawings to finding missing angles or lengths, and use of drawing instruments to duplicate or construct similar shapes: circles, squares, rectangles and triangles included. Plotting and planning routes on map may introduce location and navigation problems.

Ideally, skills and activities with actual or potential take home value clear to students, their teachers and parents would be included here for the sake of motivation and to provide a familarity with similarity practices if not principles for employment later in the development of right triangle and/or unit circle trigonometry. Right triangle trigonometry may then be introduced as a computational alternative to the careful drawing of diagrams to scale for solution of missing angles and lengths questions. The small step development of algebraic skills above could enable a faster mastery of trigonometry. So could the introduction of complex numbers and their properties.

14. A Calculus Preview and Algebraic Skill Building. But for the calculation of derivatives, the elementary application of first derivatives to graphing is one of the algebraic lighter part of calculus. Before calculus, to develop algebraic reasoning and to provide a context for the study of slopes and factored polynomials, we give students formulas for the slope of tangent lines, factored or not, for the sake of sign analysis and through that the location of intervals where a function is increasing or decreasing, as well as the location of maxima and minima inside intervals and at included endpoints.

To ease or avoid calculus shock, we may informally identify derivatives with slopes to height functions $y = f(x)$ and explain how the sign of the slope indicates where the hieght is increasing or decreasing. Then slope sign analysis may be done with formulas for slopes - derivatives - being given by factored polynomials. The underlying exercises will develop algebraic skills and provide students with an explanation of why derivatives will be calculated. Before calculus, the same lessons may provide a context for the study of slopes and polynomial.

15. Limit Evaluation Steps. Some limits can be evaluated immediately by substitution. Others requiring simplication may employed delayed substitution. Talking about immediate and delayed substitution vocalizes a common practice. The variable x is essentially a parameter in the limit definition \[ \lim_{h\to 0}\frac{f(x+h) -f(x)}{h} = f'(x)\] of derivatives. By picking three to five values for x, several numerical examples of limit evaluation may given with work done in a manner that indicates the general patterns, and sets the stage carrying the parameter x through the calculations to obtain a parameter- dependent result, and derive formula. The foregoing may ease or avoid an algebraic difficulty in calculus.

16. Error Control Perspective. The decimal-free epsilon-delta view of limits and continuity is many take years for undergraduate students in mathematics to grasp. In my college days, decimal were employed in error control questions in numerical analysis and in the illustration of limit properties, but not in the accompanying theory. The latter was decimal-free. Yet following the work of Lipman Bers Calculus text, the decimal perspective may be sufficient for student not in pure mathematics with addition of one further idea: Limited and unlimited error control. In the case of limits and continuity of functions $y=f(x)$ at $x = a$, the relevant and concrete error control question follows: To how many decimal places m must the argument x agree with a number a in order for $f(x)$ to agree with a limit L or $f(a)$ to say n decimal places. This question may have an a least answer $m = g(n)$ for some or all whole number n. In the first case, error control is limited, and the values the difference between $f(x)$ and the number $L$ cannot be controlled beyond a maximal number of n places. So jumps may occur. On the hand, if $g(n)$ is defined for all whole numbers n, error control is unlimited and we may take that as the first characterization of the existence of a limit or continuity as $x \to a$. The latter concrete perspective may serve as destination for some and a stepping to an epsilon-delta comprehension for others.

17. Deductive Logic. Deductive logic in mathematics and a model for Euclidean or Geometric reason can be introduce via mathematics-free words and examples. Logic mastery apart from and for mathematics can be emphasized as way to improve precision in reading and writing, and hence as way to avoid confusion at home, at work and in school where skills and instruction are given or followed. The notion of proof in higher mathematics is predated in arithmetic and formula evaluation by providing written work formats for doing and recording calculations in steps that can be seen and checked. Derivations and formal proofs in mathematics take the forgoing notion a little further by requiring reason for each step be clear or recorded as well, so steps and reason for them can be seen and checked. Finally, in the preparation of students for calculus, the site account of Euclidean geometry employes direct proofs only. Implication rules and proofs by contradiction are not employed. That should make the key elements of Euclidean geometry covered easier to learn and teach, and so eliminate objections to coverage of the latter in high school. The Chinese square dissection proof of the Pythagorean theorem may be not be as rigourous as other ways - some nuances are pushed under the carpet, but it is enough to add the to logical structure of mathematics as seen in school. In general, the logical structure of mathematics emerges before the formal appearance of deductive logic in the skill and concept development when the dependencies of later skills and concepts on earlier one is seen.

18. Ends and Values. At the college level, most mathematics instruction serves the needs of other disciplines. At the secondary levels, mathematics insruction may serves the needs of calculus-based college programs. But early secondary and primary school mathematics instruction may also focus on skills with actual or potential take home value clear to students, teachers and teachers. Skills with take home may be found or seen in time and date matters, money matters, arithmetic with numbers and units, geometry with maps and plans, and decision making with logic or in the presence of uncertainty. Maximally done, skills with take home value could allow and early secondary school instruction to build skills and leave student with a favourable impression of mathematics for themselves and future families, while giving a base for the further more technical studies. Such skill development may be adapted to local needs in urban, rural and aboriginal communities where completing secondary school and entering or succeeding is college is not always certain. The favourable impression has to be strong enough, so it continues even if students do not succeed in their further studies. That is the challenge. Meeting it would reduce the problem of student coming to school avoiding or disliking mathematics due to the past horrible experience of their parents

Skill Engineering Revived

18. Skill Engineering. The 21st year book The Learning and Teaching of Mathematics, Its Theory and Practice of the National Council of Teachers of Mathematics, Washington D. C. 1953, reflects a Rip Van Winkle direction of for endeavours in mathematics education:

  • page 349. .. in a sense the teacher must be a master technician. He must know how to build any known kind of learning. .. must weigh, balance, and appraise the possible learning. ... know their relative worth both for the individual and for society.

  • page 348. a teacher is a learning engineer, a builder of minds that will solve problems. As such, he must first know the total mathematics he will teach, that is, the framework.

  • page 248. There are some persons who say one who knows cannot teach for he cannot fathom the difficulties of his students. These persons say that as a teacher work with his students through a problematic situation which is new to both teacher and student, real learning takes place and then only. We believe this assumption to be entirely erroneous and assert that a teacher is a learning engineer ...

The quotes are not taken out of their 1953 context. However, they may be inconsistent with the constructivist inclinations of the NCTM since 1989. Skills that can be seen, can be confirmed, corrected and practiced as needed. Observable skills are tangible. Skill development paths may be subject critical path analysis and optimization in accordance with ends and values, including inductive criteria which calls for base steps in each course to be within the reach of students, and each further step to be within reach of students after mastery of earlier skills. Here steps missing, too large or skipped may lead skill engineering to falter. Profound student learning difficulties may have the same effect. That being said, mathematics education in primary and secondary schools may seek a vulgarization of skill development which provides ends, values and methods for skill development strong and simple enough to guide instruction, even when instructor need training in mathematics. The description of mathematics programs in terms that people with doctorates in matheamtics cannot follow needs to be replace by description and materials, all in plain language that elementary and secondary teachers can follow and apply. That being said, No single academic or educational authority should lead or dominate. schools or school districts should be able to choose between competing mathematics education programs they will follow. Then natural selection may lead to competition and continuous improvement of programs. And if a monopoly or duopoly is seen, alternative programs should be created with the mission of providing an alernative paths.


Revised December 11, 2011.



www.whyslopes.com >> Mathematics Skill Development Framework >> Multiple Ways to Improve Mathematics Skill Development Next: [Ends Values Methods For Skill Development - Framework Prequel.] Previous: [ Phase 5. Calculus Light to Rigourous for 1 to 2 years.]   [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13][14] [15] [16] [17]

Return to Page Top

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?

Play with this [unsigned] Complex Number Java Applet to visually do complex number arithmetic with polar and Cartesian coordinates and with the head-to-tail addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.

Pattern Based Reason

Online Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule- and pattern-based reason and decisions in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not reach it. Online postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge.

Site Reviews

1996 - 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. ...

Senior High School Geometry

- Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.
- Complex Numbers - Learn how rectangular and polar coordinates may be used for adding, multiplying and reflecting points in the plane, in a manner known since the 1840s for representing and demystifying "imaginary" numbers, and in a manner that provides a quicker, mathematically correct, path for defining "circular" trigonometric functions for all angles, not just acute ones, and easily obtaining their properties. Students of vectors in the plane may appreciate the complex number development of trig-formulas for dot- and cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no knowledge and some knowledge of the tangent function in trigonometry.

Calculus Starter Lessons

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.
- Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima and minima while indicating why we calculate derivatives or slopes to linear and nonlinear curves y =f(x)
- Arithmetic Exercises with hints of algebra. - Answers are given. If there are many differences between your answers and those online, hire a tutor, one has done very well in a full year of calculus to correct your work. You may be worse than you think.


Return to Page Top