Title: THE TRANSITION FROM ARITHMETIC TO ALGEBRA: WHAT WE KNOW AND WHAT WE DO NOT KNOW (Some ways of asking questions about this transition)?
1THE TRANSITION FROM ARITHMETIC TO ALGEBRA WHAT
WE KNOW AND WHAT WE DO NOT KNOW(Some ways of
asking questions about this transition)?
2This talk will be short on
- Time
- Concrete Classroom Examples
3This talk addresses algebra as a mathematical
discipline, and its relation to the course(s) in
high school labeled algebra.
4BUT 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.
5So 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.
6What 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
7What 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
8What 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.
9This 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.
10THE FIRST LEARNING TRAJECTORY THREE WAYS TO
THINK ABOUT ALGEBRA
- As the general arithmetick
- Newton
- coming
- coming
11A Algebra as the general arithmetic
- 25 5x5
- 24 6x4
- 49 7x7
- 48 8x6
- Etc., so
- A2-1 (A1)(A-1)?
12Sample 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.
14THE FIRST LEARNING TRAJECTORY THREE WAYS TO
THINK ABOUT ALGEBRA
- A. As the general arithmetic
- B. As the study of binary operations
- C. coming
15Contrast
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.
16These 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.
18Assertion III.
- Students who are solving equations algebraically
(and not arithmetically) are already working
algebraically, using general properties of binary
operations.
19Key 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
20THE 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.
24Key teaching questions for level C
- What plays the role of A?
- What plays the role of B?
25A SECOND LEARNING TRAJECTORY TWO TYPES OF
REASONING
- Inductive reasoning from the specific to the
general - Deductive reasoning from the general to the
specific.
26Inductive Reasoning
- Describing patterns
- Making conjectures
- Testing hypotheses
- Passing from specific cases to general rules
27Deductive 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
28Obviously.
- 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.
29Assertion IV
- Students making the transition from arithmetic
to algebra are typically focused on learning and
applying inductive reasoning, rather than
deductive reasoning.
30WHAT ABOUT THE DISTRIBUTIVE LAW? ISNTHAT AN
AXIOM?
31WHAT 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.
32Assertion 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.)?
33SO
- 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?
34ASSERTION VI
- Traditionally, in school mathematics
- Algebra is thought of in connection with
inductive reasoning - Geometry is thought of in connection with
deductive reasoning.
35QUESTIONS
- 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?
36QUESTIONS
- 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?
37QUESTIONS
- 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?