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.
-
Scalar multiplication by $a$ distributes over addition. Here
multiplication by $(r_1, 0)$ is equivalent to a scalar
multiplication.
-
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.
|
|