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 II
B. 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]
B: Algebra as the study
of binary operations
Contrast:
Solution II:
Solve 2x+5 = 13.
Solution I:
2x1+5=7, too small
2x2+5=0, too small
2x6+5=17, too big
2x4+5 = 13 just right
so x = 4.
2x + 5 = 13
subtract 5 from each
side:
2x = 13 - 5 = 8
Divide each side by
2:
x = 4.
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; then
4x2-1 = (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?
WHAT ABOUT THE DISTRIBUTIVE LAW?
ISN’THAT AN AXIOM?
Well, yes, but:
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.
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?