THE TRANSITION FROM ARITHMETIC TO ALGEBRA:

WHAT WE KNOW AND WHAT WE DO NOT KNOW

(Some ways of asking questions about this transition)

This talk will be short on

– Time

– Concrete Classroom Examples

This talk addresses ‘algebra’ as a mathematical discipline, and its relation to the course(s) in high school labeled ‘algebra’.

BUT WHAT IS ALGEBRA?

• I will not give a satisfying answer to this question.

• I will give constraints, valid for this talk only, on what I’m referring to by the term ‘algebra’.

• Only a fool will answer a question easily that 1000 wise people have asked.

So I’m leaving out, here and now, the following very important topics

(among others):• Graphing equations and inequalities• Translating words to symbols• Computations with complex numbers• Computations with radicals• Applications• Motivations• Etc., etc.

What Algebra is NOT: Part I

A: Algebra is not the study of letters used in place of numbers

5 + what? = 12“My rule is take a number and add 7”“My rule is [ ] -----> [ ]+7”“My rule is x ------> x+7”

What algebra is NOT: Part IIB. Algebra is NOT characterized by the study of

functions

>Graphing functions leads to analysis, not algebra

>There are many ways to represent functions…

>…some of these representations are algebraic,

BUT there is more to for algebra than just the representation of functions

What about the ‘functions approach’ to algebra?

Assertion I: The heart of algebra is NOT an understanding of the function concept. (Algebra deals with the study of binary operations.)

BUT: In looking at functions, and the role of algebraic variables in representing functions, students can come to understand something about binary operations.

This is not unusual:>Geometry is NOT characterized by the use of an axiomatic

system.

BUT: In studying geometry, students can come to understand something about the use of an axiomatic system.

>Fractions are NOT characterized by the expression of the probability of an event.

BUT: In computing probabilities, students can come to understand something about fractions.

>Baseball is not characterized by the speed at which a player runs.

BUT in playing baseball, one can increase one’s ability to run quickly.

THE FIRST LEARNING TRAJECTORY:

THREE WAYS TO THINK ABOUT ALGEBRA

–As “the general arithmetick”[Newton]

–[coming]

–[coming]

A: Algebra as ‘the general arithmetic”

25 = 5x5

24 = 6x4

49 = 7x7

48 = 8x6

Etc., so

A2-1= (A+1)(A-1)

Sample teaching questions for level A:

What is the next step in the pattern?

What is the 1000th step in the pattern?

What is the 1001st step in the pattern?

Assertion II: Students transitioning from arithmetic to

algebra are learning to generalize their knowledge of the arithmetic of rational numbers.

Alternatively:Students transitioning from arithmetic to

algebra are working on the level of algebra as ‘the general arithmetic’.

THE FIRST LEARNING TRAJECTORY:

THREE WAYS TO THINK ABOUT ALGEBRA

A. As “the general arithmetic”

B. As the study of binary operations

C. [coming]

Contrast: Solve 2x+5 = 13.Solution I: 2x1+5=7, too small2x2+5=0, too small2x6+5=17, too big2x4+5 = 13 just rightso x = 4.

Solution II:

2x + 5 = 13

subtract 5 from each side:

2x = 13 - 5 = 8

Divide each side by 2: x = 4.

B: Algebra as the study of binary operations

These are all the same for student II, but not for student I:

2x + 5 = 13

2x + 5 = 12

2756x + 593 = 1028

.35x + .2 = 1.7

2/3 x + 4/5 = 7/8

Etc.

On this level:

• Students begin thinking of binary operations, and not just the numbers the operations are applied to, as objects of study.

• “Thinking about computations” happens on this level, or is a hallmark of this level of work.

• The “-tive laws” (commutative, associative, etc.) begin to have real meaning on this level.

• Algebra as the study of ‘structures’ becomes possible.

Assertion III.

Students who are solving equations algebraically (and not arithmetically) are [already] working algebraically, using general properties of binary operations.

Key teaching questions for level B:

“How are these equations the same?”“What do you do next?” [i.e. before the

students has actually done a computation]

“What do you want to do with the calculator?” [i.e. before the student has picked it up]

THE FIRST LEARNING TRAJECTORY:

THREE WAYS TO THINK ABOUT ALGEBRA

A. As “the general arithmetic”

B. As the study of binary operations

C. As the study of the ‘arithmetic’ of the field of rational expressions.

“In arithmetic we can use letters to stand for numbers. In algebra, we use letters to stand for other letters.”

--I. M. Gelfand

• A2-B2= (A+B)(A-B)• Let A = 2x; B = 1; then4x2-1 = (2x+1)(2x-1)• Let A = cos x; B = sin x;cos2x – sin2x = (cos x + sin x) (cos x – sin x)

(the last example is not strictly about rational expressions…)

On this level:

• The form of algebraic expressions becomes important

• Students can develop an intuition about which of several equivalent forms is the most useful for a given situation

• Algebraic expressions become objects of study, and not just their value at a given ‘point’.

Key teaching questions for level C:

“What plays the role of A?”

“What plays the role of B?”

A SECOND LEARNING TRAJECTORY: TWO TYPES OF

REASONING• Inductive reasoning: from the specific to

the general

• Deductive reasoning: from the general to the specific.

Inductive Reasoning

Describing patterns

Making conjectures

Testing hypotheses

Passing from specific cases to general rules

Deductive Reasoning

Examining assumptions

Making definitions

“Proving theorems” (I.e. linking the truth of one statement to the truth of another)

Passing from general rules to specific cases

“Obviously….”

…often means that a statement is recognized by the speaker to be true because it is derived from another statement, rather than because the speaker has observed it to be true.

“Obviously, if you’ve crossed a bridge you’re not in Manhattan any more.”

Assertion IV

Students making the transition from arithmetic to algebra are typically focused on learning and applying inductive reasoning, rather than deductive reasoning.

WHAT ABOUT THE DISTRIBUTIVE LAW?

ISN’THAT AN AXIOM?

Assertion V:

“Applying the distributive law” in a computation is, for us, an example of deductive reasoning.

But for most students, most of the time, it is only deductive reasoning *after* they’ve recognized deductions in other contexts.

WHAT ABOUT THE DISTRIBUTIVE LAW? ISN’THAT AN AXIOM?Well, yes, but:

Assertion V

“Justification of computation” is not a very effective step in learning about deduction.

BUT if this is done within a very conscious framework of, say, the field axioms, it can be a good example of a deductive system.

(This is an empirical statement, made on the basis of experience.)

SO:

• How do we support students learning about the special nature of mathematical truth?

• What are their typical intuitions about deductive logic?

• What are the steps in the development of this concept that we can anticipate them passing through?

ASSERTION VI:

Traditionally, in school mathematics:

Algebra is thought of in connection with inductive reasoning;

Geometry is thought of in connection with deductive reasoning.

QUESTIONS

1. How true is assertion VI?

Are there places in algebra where we develop of deductive reasoning?

Are there places in geometry where we develop inductive reasoning?

QUESTIONS

2. How true ‘ought’ Assertion VI to be?

Is there a reason that algebra is more conducive to inductive reasoning and geometry to deductive reasoning?

Should we take opportunities to make Assertion VI ‘less true’?

QUESTIONS

3. How do we help students progress from inductive to deductive reasoning?

4. Or is ‘progress’ the wrong word for the relationship between the way we learn about these two processes?