Chapter 10: Explaining Algebra

Previous: 1 Two Barriers- Words Before Symbols, Symbols Better Seen and Read Silently than Read Aloud

Following a mastery of counting, arithmetic and the use of simple formulas, the algebraic way of writing and reasoning can be introduced by talking about and then illustrating the following three skills for algebra:

1. We can describe numbers and quantities.

2. We can describe calculations that are done or might be done. This description can be done with words or with a symbolic (algebraic) shorthand notation, that is formulas.

3. We can change how a number or quantity is computed. Here symbolically described rules of arithmetic (properties of real numbers) say how.

Talking about these three skills compensates say for the nonverbal nature of the symbolic or algebraic description of formulas – better seen and read silently than spoken aloud. The initial discussion and illustration of the three skills should show and emphasize that there are two notions of variables, one with and one without symbols, that there is more to mathematics than just doing arithmetic; and that the symbolic or algebraic description of calculations is more compact than a word description. The algebraic description however requires an understanding or definition of the symbols involved – their roles. And as motivation for algebra, instructors may observe to students that the main service of this algebraic notation is to describe calculations or help change the way that they are done in all numerical disciplines. Discussing and illustrating the three skills also provides a mechanism for alleviating math phobia[2].

How to Present the Skills

First Skill: Talking about Numbers and Quantities. We can talk about numbers and quantities and, in doing so employ the everyday meanings of the adjectives constant, variable, known, unknown, given, forgotten, .. in the description of numbers and quantities. Some words or examples may be needed to show the difference between numbers and quantities. The latter involves units of measurement. Use of units and calculations involving quantities needs to be sanctioned, since we are not necessarily concerned with treating math as solely dealing with numbers. Quantities appear in every day life and many computational disciplines, economic, technological or scientific.

Second Skill: Describing Calculations. We can describe calculations that may be done or postponed or never done. This description can be done with words alone or with formulas, that is algebraic shorthand notation. A few words and examples are appropriate to show the equivalence in some simple cases and to show the advantages of the shorthand description in other cases. The aim here is to emphasis that advantage, while mentioning the cost: algebraic formulas are better seen and read silently. The latter has been a nonverbal obstacle to the discussion of algebra and algebraically (or symbolically) derived and described results in mathematics.

When students are not yet familiar with complicated formulas, that is calculations hard to describe with words alone, the lessor aim is just to give the following understanding: formulas for numbers or quantities are simply symbolic or algebraic ways of describing calculations that might be done. Then when complicated formulas appear, the tongue in cheek exercise of describing them with words only can be given. Language teachers in the later years of secondary school can give students the essay option of clearly describing the evaluation of the calculations encoded and represented by the quadratic formula or the compound interest formula.

Third Skill: Changing Calculations. We can change the way a number or quantity is computed. Here students may be familiar with cases where two different calculations give the same result. In such situations, one computation, preferably the simplest, can replace the other to lessen the amount of computation. The arithmetic review problems [3] contain a few examples or hints of such computational shortcuts. A further motivation is the message: formulas for numbers and quantities may be obtained from other formulas by replacing one computation by another that gives the same result, or by interchanging a calculation with a symbol that represents its result. The interchange here is possible in two ways. Students can be given examples to support this message.

Describing and Changing Calculations

Three topics may further introduce and illustrate the algebraic way of writing and thinking. Given the difficulty that many have had in mastering IT, saying too much or redundancy in the explanation is better than saying too little. The algebraic way of writing and thinking is not obvious. It involves some rationalizations which the naturally adept may acquire via osmosis, but which others, artificially adept, can understand as well. The topics are as follows.

1.  Area and Volume Calculations – the substitution concept. Here two formulas for the volume of a box are shown to be interchangeable. Here the area formula A = WL is solved for L and W. See below. 

3. The solution of linear equations with one, two and more unknowns using numerical or symbolic coefficients. Here there is a message that sometimes working with the numerical coefficients is better – less involved. Here numbers go from being unknown to known. The expression find the unknowns could be better rephrased make the unknowns, known.

These three topics provide opportunities to show and illustrate various substitution and replacement ideas or laws. Their presentation and discussion at length following the three skills should I conjecture give a mastery of the algebraic way of writing and reasoning sufficient to understand the symbolic and algebraic statement of the properties of real numbers.

Area and Volume Calculations

These first illustrations from the book Three Skills for Algebra are based on the calculation of the area A of a rectangle of width W and length L. These examples introduce the replacement and substitution operations in settings that should be accessible and familiar to students who have mastered in primary school, the use of simple formulas.

In shorthand notation, A = W L. This gives a means for computing A from the other two quantities. Next I note that W = [(WL)/(L)] due to the multiplication reversal by division rule. And then the equality WL = A suggests W = [(A)/(L)]. This provides a first example of algebraic reasoning. It based on the multiplication reversal by division rule and the replacement of WL by its equal A. The letter A can be thought of as shorthand for the result of W times L. Solving for L provides an another example and hints at the interchangeability of the roles of length L and width W. For mathematical novices, that is, students, I would not invoke this interchangeability or symmetry. It would be an afterthought - a comparison of two similar computations and a suggestion that we could have used a symmetry argument.

The second illustration is the calculation of the volume V of a box of height H whose base has area A, width W and length L. The volume of the box is V = H ×W×L, where the order of the product and the grouping in it does not matter. Students can be asked to verify the latter for an example or two. The grouping V = H (WL) implies V = HA by the replacement (again) of WL by its equal A. So there are two very different ways to compute the volume: one using H and A and the other using H, L and W. The reverse replacement, that of A by WL in the formula V = HA yields V = H (WL). This is another example of algebraic reasoning: it reinforces the replacement idea.

Before or during both of the above examples, students can be told the following verbal rule: multiplying a first number by a second nonzero number and then dividing the product by the second, yields the first example. A few examples can confirm it. This multiplication reversal by division rule can be given in verbal form. Remember the students are still perhaps prealgebraic.

Before or during both of the above examples, students can be told the following verbal rule: multiplying a first number by a second nonzero number and then dividing the product by the second, yields the first example. A few examples can confirm it. This multiplication reversal by division rule can be given in verbal form. Remember the students are still perhaps prealgebraic.

Interest Formula Examples

A third illustration employs the compound interest (or investment) formula. (The simple interest formula could be used instead). Here the derivation or justification of the formula in the work may be done before and/or after this illustration – repetition here is not harmful, and there may be alternative viewpoints.

The compound interest formula A = P (1+i)ⁿ can be employed to compute the final amount A from a knowledge of the principal P, the interest rate i, and the number of periods n invested. But the formula can also be used to solve for

P = P (1+i)n (1+i)n = A (1+i)n

using the multiplication reversal by division rule. Do this once with numbers given for A, i and n, and once with the letters instead to show how the arithmetic and algebraic patterns agree, but that one is more general. In the arithmetic solution, I postpone all evaluations of arithmetic operations and all simplification of fractions so that the arithmetic-algebraic pattern is more obvious.

After a discussion of powers and roots, the compound interest formula A = P (1 +i)ⁿ can be used to obtain the value of i in the event that numbers are given for the other quantities, or a formula for the interest i in the general case. Here again, one may solve an arithmetic problem in a manner that resembles the more general solution. The latter is given second.

These manipulations of the compound interest formula further lead students to an algebraic way of thinking. They show that the algebraic way has the potential to give a formula or pattern to solve many similar problems at once. To this end, I may insist on following all the steps in an arithmetic problem, exactly as in the algebraic solution, until students impatiently suggest that we use the algebraic formula (and thereby omit the chain of reasoning that lead to it). This marks a turning point in their comprehension.

The foregoing represents an inductive and a psychological approach to the explanation and comprehension of the algebraic way of writing and thinking. What is missing now are examples to reinforce it, and rules formally stated to say when two different calculations give the same result.

Linear Equation Examples

Solving linear equations provides a further confidence building component for the algebraic way of writing and reasoning. Here to a first number adding and then subtracting a second yields the first. Physically this echoes the notion that adding and then subtracting the same number of marbles to a bag of marbles leaves the count of marbles in the bag equal to its original value. The algebraic pattern is (a+b)-b=a This be can confirmed with a few examples (with b positive if students are not yet familiar with the subtraction of negative numbers).

Next containers for numbers can be labelled with letters, and the letters used as shorthand symbols or abbreviations for the contents. This allows us to speak of numbers or quantities without them having much physical or economic significance.

In explaining the solution of linear equations, we may start with several numerical examples of one equation with one unknown, for instance, 5x+7=28, and then solve them in following the algebraic pattern used to solve ax+b=c. Following the algebraic pattern means that arithmetic operations involving the coefficients are recorded but not done nor simplified, even if that tries the patience of student. The numerical examples again lead to and corroborate the algebraic solution. The latter can be derived following the same pattern or reasoning process employed in the postponed arithmetic examples. The solution formula x = (c-b)/a is seen to solve many similar problems at once. This offers more incentive for the algebraic way of writing and thinking.

Following one equation with one unknown, we may offer numerical of triangular systems with two to several unknowns. These triangular systems can be solved in a forward or backward substitution manner. (The term triangular stems from the location of the nonzero coefficients in matrix representations of such systems). Their solution builds confidence and puts students at ease with working with several unknowns. Students can be told or shown that the algebraic pattern becomes complicated, and that arithmetic approach requires less work than obtaining a formula – a limitation on the algebraic way of writing and reasoning has appeared. With the triangular systems, we may include systems of equations equivalent to a triangular system after a change in order of the equations. This presents a slight variation on the theme of backward or forward substitution for solving lower and upper triangular systems.

Next, the reduction of general linear systems to triangular systems to solve them, can be shown for two and more unknown numbers.

At some point in this solution process, the idea of checking results can be emphasized. The solution of linear systems follows long chains of reason prone to error.

The foregoing steps increase the confidence of students and makes them at ease with looking for numbers that are initially unknown. Solving a consistent system of equations can be characterized as changing the psychological state or knowledge of some numbers or quantities from unknown to known.

Calling the unknowns in a linear equation variables is a somewhat objectional, yet presently standard abuse of language. The frequently employed letters x and y are used to represent the numbers, known or not, in a linear system. They may be called variable if it is accepted that for a given linear system, they represent fixed numbers in the solution of the system, but they and the solution may change or vary from system to system. Use of the term variable should be justified, otherwise the use is an abuse of language, a too common one.

Arithmetic Rules for Algebra

After some or all of the previous topics, students should be thinking algebraically. These topics, together with a nascent algebraic thinking, provide a context for the comprehension of the algebraically described properties of real number arithmetic. Most can be introduced or described as assumptions or rules which say when two different computations give the same result. Calling an assumption an axiom or law may disguise its humble origin: an assumption is an assumption is an assumption.

In the companion book Three Skills for Algebra, the chapter Arithmetic Rules for Algebra, henceforth called the chapter, illustrates the computational significance of each rule, that is it provides an interpretation for each one.  For instance, the commutative law for multiplication represents the idea that the order of the factors does not affect the results. It can be stated for a pair of real factors or several. The rule for several can be derived, if one wants to complicate matters, from the rule for a pair. The chapter omits the rule for several (that could be rectified) but it does illustrate the rule for a pair with decimal numbers and it emphasizes that the factors could be given by the results of one or two formulas.

The chapter emphasizes that the properties described algebraically and symbolically, imply methods or rules for changing the way calculations are done as well methods for simplifying arithmetic. And in both, substitution may be employed.

The chapter also emphasize the while the state laws involve only addition and multiplication, laws of arithmetic for subtraction and division follow as subtraction can be regarded as the addition of a negative (additive inverse) and as division can be recast in terms of multiplication by the reciprocal (multiplicative inverse).

Units in Computation

The exposition of mathematics in secondary school should acknowledge, support and sanction this computational role of mathematics in other disciplines. The chapter also emphasizes that the properties of real numbers also apply to quantities – that is, real numbers times units of measurement for weight, mass, speed, distance, time, temperature or monetary amounts, etc. Currently mathematics courses, except examples in trigonometry, only discuss real numbers and forgo or avoid any discussion of the units or quantities that appear in science, technology or commerce.

It is possible to formulate all computations without units, but without any extra work, units can be carried through computations algebraically or symbolically in the same manner as indeterminates. For the sake of algebra across the curriculum, the sanction in mathematics courses of units in computations is recommended. Otherwise, there is a void. Mathematics curricula need to sanction and teach the algebraic abilities required by other subjects.

Implication Rules

When mathematics is only described and not derived, implication rules and logic are missed or not noticed. It is possible to identify them or at least the presence of some reasoning process. Every use of the terms or phrases such as therefore, thus, hence, and from this signals the drawing of a conclusion. Further, any multistage rule-based process which yields a result or conclusion gives an example of a chain of reason. Beyond this, the appearance of implication rules and their contrapositives in the statement of axioms or assumptions can be brought to the attention of students. The virtual absence of synthetic geometry in schools and college makes this extra attention to implication rules (conditional statements) and their contrapositive necessary. Reminders follow.

If a and b are both positive numbers then their product ab is nonzero.

If a and b are both nonzero real numbers then their product ab is nonzero.

a and b are both nonzero real numbers

The quadratic formula can be derived by completing the square, or by showing the expansion of

2. A discussion of unlimited accuracy in computations leads to concepts of continuity and limits. Limits on accuracy lead to the concepts of discontinuity in functions or their machine based computations. See the chapter Limits, Error Control and Continuity in the companion book Why Slopes and More Math. Gifted students in high school should able to follow and discuss the error control perspective of continuity, and its contrapositive formulation. Otherwise this topic provides college level material.

[2] Primary school math instruction corresponds to teaching children how to wade in a paddling pool not deep enough for swimming. Post primary math instruction corresponds to teaching students to swim, that is to employ deductive reason and algebraic ways of writing and thinking. Both appear and are required after arithmetic in mathematics.

Swim instruction today favours non-swimmers entering the shallow end of a pool to bounce up and down while exploring and practicing swimming motions. Many of these non-swimmers then gain a dynamic sense of balance or buoyancy and so begin to swim. Following this, their technique or strokes can be refined.

Learning how to swim by wading and bouncing about in the shallow end is not always successful but it may be more successful and encouraging than the old fashioned approach of getting students to jump in the deep end. This old-fashioned approach leads to the identification of people with a natural talent for swimming. Yet it immediately discourages others and leads them to the notion that they have no natural talent for swimming. So it should be avoided. Yet besides people with a natural talent, thus recognized, there are others with a cultivatable talent, those who can learn to swim by wading in the shallow end to gradually practice swimming motions in the hope of obtaining a (dynamic) sense of buoyancy.

There may be a similar situation in mathematics. The presentation and illustration of the three skills for algebra in particular give another approach for cultivating the mastery of algebraic writing and reasoning skills while avoiding the perils and phobias of sudden immersion.

[3] in the companion work Three Skills for Algebra

[4] The use of the compound interest formula assumes that students are familiar with it – simpler examples to show the importance of solving for certain quantities could be based on the simple interest formula I=Pit or with the formula A=P(1+it).

[5] The discussion of how this state changes when the system is inconsistent is left to the reader :)

[6] In retrospect, the chapter should called Symbolically or Algebraically Described Rules of Arithmetic.

[7] Note again, intermediate level instruction in the 1960s or 70s modern math curriculum did not acknowledge the decimals representation of real numbers and provides no alternative. Here real numbers of great interest and service were not immediately defined in that their decimal representation is not sanctioned (nor fully exploited). Following these modern math curricula, the precise set-theoretic definition of real numbers is left to a college courses seen by very, very few students. All others are left in suspense.

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Volume 1B, Mathematics Curriculum Notes,

 Foreword + Chapters 1 to 10 + 12

Area Map
Inductive Principles
1 Introduction
2 For & Against Math
3 Algebraic Thought
4 Why Slopes & SQRT of -1
5  Books & Articles to Read
6 Unruly Origins of Algebra
6. Axiomatic Civilization
7 Geometry, 2 Ways
8 Modern Instruction
9 The Two Ends
10 The Transition
10 Explaining Logic
10 Explaining Algebra
10 Why Sets in Math.
12 Four Phases
Links

Chapter 11: Primary School Mathematics

11 Primary Math
11 Cue Cards
11 Counting
11 Decimals - Addition
11 Decimals -Times
11 Decimals & Subtraction
11 Fractions and Division
11 Notational Conflict
11 Reciprocals Etc
11 Decimals - Ratios
11 Size Comparison
11 Numbers, +ve or -ve
11 Rename < Sign
11 Complex Numbers

Will provide an alternative to Chapter 11 later, most likely in the Parent's Area: Help Your Child or Teen Learn 

Most students in high school are not heading for calculus, but most topics in high school mathematics are present due to calculus.  Preparation for calculus demands their coverage at  full strength.

See too, this site 55+,  Math Education Essays. Site areas and pages provide pieces of the a Mathematics Education, Jigsaw Puzzle, in formation.

-Inductive principles for course design & delivery  require a clear description of where and how skills and concepts may rest on earlier ones, so that difficulties may be explained and remedied by looking for  what was missed or forgotten in earlier studies. 

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Mathematics is a demanding subject. All errors in notation and comprehension need to be identified and corrected. In reading, spelling and writing, students have to learn all the letters in the alphabet, not just some. and memorize spelling. Anything less implies difficulty.

Likewise in mathematics, students have to master key skills and concepts, one at a time and one after another. Anything less implies difficulty.

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Modern mathematics curricula introduced an inconsistency into course design and delivery. They did not sanction the use of decimals nor the use of diagrams in skill and concept development but decimal arithmetic and diagrams are needed for student comprehension and for an operational mastery of quantitative skills. That implies the need for an mixed-math curricula based on a systematic development of operational skills, sufficient for applications and sufficient to provide a base & context  for  the very optional study of pure mathematis.

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