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THE TRANSITION FROM ARITHMETIC TO ALGEBRA: WHAT WE KNOW AND WHAT WE DO NOT KNOW (Some ways of asking questions about this transition)?

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Title: THE TRANSITION FROM ARITHMETIC TO ALGEBRA: WHAT WE KNOW AND WHAT WE DO NOT KNOW (Some ways of asking questions about this transition)?


1
THE TRANSITION FROM ARITHMETIC TO ALGEBRA WHAT
WE KNOW AND WHAT WE DO NOT KNOW(Some ways of
asking questions about this transition)?
2
This talk will be short on
  • Time
  • Concrete Classroom Examples

3
This talk addresses algebra as a mathematical
discipline, and its relation to the course(s) in
high school labeled algebra.
4
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 Im referring to by the term
    algebra.
  • Only a fool will answer a question easily that
    1000 wise people have asked.

5
So Im 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.

6
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 -----gt 7
  • My rule is x ------gt x7

7
What algebra is NOT Part II
  • B. Algebra is NOT characterized by the study of
    functions
  • gtGraphing functions leads to analysis, not
    algebra
  • gtThere are many ways to represent functions
  • gtsome of these representations are algebraic,
  • BUT there is more to for algebra than just the
    representation of functions

8
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.

9
This is not unusual
  • gtGeometry 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.
  • gtFractions are NOT characterized by the
    expression of the probability of an event.
  • BUT In computing probabilities, students can
    come to understand something about fractions.
  • gtBaseball is not characterized by the speed at
    which a player runs.
  • BUT in playing baseball, one can increase ones
    ability to run quickly.

10
THE FIRST LEARNING TRAJECTORY THREE WAYS TO
THINK ABOUT ALGEBRA
  • As the general arithmetick
  • Newton
  • coming
  • coming

11
A Algebra as the general arithmetic
  • 25 5x5
  • 24 6x4
  • 49 7x7
  • 48 8x6
  • Etc., so
  • A2-1 (A1)(A-1)?

12
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?

13
  • 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.

14
THE FIRST LEARNING TRAJECTORY THREE WAYS TO
THINK ABOUT ALGEBRA
  • A. As the general arithmetic
  • B. As the study of binary operations
  • C. coming

15
Contrast
B Algebra as the study of binary operations
Solution II 2x 5 13 subtract 5 from each
side 2x 13 - 5 8 Divide each side
by 2 x 4.
  • Solve 2x5 13.
  • Solution I
  • 2x157, too small
  • 2x250, too small
  • 2x6517, too big
  • 2x45 13 just right
  • so x 4.

16
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.

17
  • 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.

18
Assertion III.
  • Students who are solving equations algebraically
    (and not arithmetically) are already working
    algebraically, using general properties of binary
    operations.

19
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

20
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.

21
  • In arithmetic we can use letters to stand for
    numbers. In algebra, we use letters to stand for
    other letters.
  • --I. M. Gelfand

22
  • A2-B2 (AB)(A-B)?
  • Let A 2x B 1 then
  • 4x2-1 (2x1)(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)?


23
  • 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.

24
Key teaching questions for level C
  • What plays the role of A?
  • What plays the role of B?

25
A SECOND LEARNING TRAJECTORY TWO TYPES OF
REASONING
  • Inductive reasoning from the specific to the
    general
  • Deductive reasoning from the general to the
    specific.

26
Inductive Reasoning
  • Describing patterns
  • Making conjectures
  • Testing hypotheses
  • Passing from specific cases to general rules


27
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

28
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 youve crossed a bridge youre not
    in Manhattan any more.

29
Assertion IV
  • Students making the transition from arithmetic
    to algebra are typically focused on learning and
    applying inductive reasoning, rather than
    deductive reasoning.

30
WHAT ABOUT THE DISTRIBUTIVE LAW? ISNTHAT AN
AXIOM?
31
WHAT ABOUT THE DISTRIBUTIVE LAW? ISNTHAT 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 theyve
    recognized deductions in other contexts.

32
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.)?

33
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?

34
ASSERTION VI
  • Traditionally, in school mathematics
  • Algebra is thought of in connection with
    inductive reasoning
  • Geometry is thought of in connection with
    deductive reasoning.

35
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?

36
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?

37
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?
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