some essential algebraic ways of thinking for success in (beginning) collegiate mathematics guershon...
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Some Essential Algebraic Ways of Thinking for success in (beginning) collegiate mathematics
Guershon HarelUniversity of California, San Diego
http://www.math.ucsd.edu/~harel
Piaget, J., Inhelder, B., & Szeminska, A. (1960). The Child’s Conception of Geometry.
“As often happens in psycho-genetic development, a mental operation is deceptively simple when it has reached its final equilibrium, but its genesis is very much more complex.”
Some Essential Algebraic Ways of Thinking for success in (beginning) collegiate mathematics
1. Thinking in terms of functions2. Representing concepts, statements,
and problems algebraically3. Symbolic manipulation skill4. Structural reasoning5. Expressing algebraically ALL the
problem constraints6. Definitional reasoning7. Geometric thinking
Some Essential Algebraic Ways of Thinking
1. Thinking in terms of functionsJack and Jill run 10 kilometers. They start at the same point, run 5 kilometers up a hill, and return to the starting point by the same route. Jack has a 10 minute head-start and runs at the rate of 15 km/hr uphill and 20 km/hr downhill. Jill runs 16 km/hr uphill and 22 km/hr downhill. How far from the top of the hill are they meet? What is the distance between Jack and Jill at any given moment from the time Jill leaves until Jack arrives?
Some Essential Algebraic Ways of Thinking
2. Representing concepts, statements, and problems algebraically
• v is in the span of u1 and u2 • c is an eigen value of A• u1 and u2 are linearly independent• The parabola is a symmetric figure• If the second differences of a pattern are constant,
then the pattern is quadratic
Some Essential Algebraic Ways of Thinking
3. Symbolic manipulation skill
Algebraic invarianceAlgebraic invariance is the way of thinking where one recognizes that algebraic expressions are manipulated not haphazardly but with the purpose of forming a desired structure while maintaining certain properties of the expression invariant.
2 0 ( 0)ax bx c a 2( )x T L
0.14 12.91 14 1291
12.91 12.91 100 1291
0.14 0.14 100 14
Proportional reasoning
Understanding decimals
tan d 1duu
Some Essential Algebraic Ways of Thinking
4. Structural reasoningoperative thought (Piaget)
The set of the common points between a line and a conic section is either empty, consists of one point, or consists of two points.
0
( )lim
n n
h
x h x
h
Is there a number larger than each term of the sequence ?2, 2 2 , 2 2 2 ,
2 is less than 2
Therefore
2 2 is less than 4
Therefore
2 2 is less than 2
Therefore
2 2 2 is less than 4
Therefore
2 2 2 is less than 2
2 1.41421 2
2 2 1. 8478 2
2 2 2 1. 9616 2
Therefore
Every term in the
sequence is less than 2.
Some Essential Algebraic Ways of Thinking
5. Referential symbolic reasoning
Non-referential symbolic reasoning refers to the behavior of operating on symbols as if they possess a life of their own, not as representations of entities in a coherent reality
With this way of thinking, one does not attend to meaning.
Some Essential Algebraic Ways of Thinking
• Non-referential symbolic reasoning(loga+logb)/logc=(a+b)/c
Row reduction preserves row-spaceRow reduction preserves column-space
(x,y) on C1 transformed into (x+a,y) on C2.If C1: y=f(x), then C2: y=f(x+a)
Some Essential Algebraic Ways of Thinking
6. Expressing algebraically ALL the problem constraints
Rectangular Land ProblemA farmer owns a rectangular piece of land. The land is divided into four rectangular pieces, known as Region A, Region B, Region C, and Region D, as in the figure:
One day the farmer’s daughter, Nancy, asked him, what is the area of our land? The father replied:
I will only tell you that the area of Region B is 200 m2 larger than the area of Region A; the area of Region C is 400 m2 larger than the area of Region B; and the area of Region D is 800 m2 larger than area of Region C.
What answer to her question will Nancy derive from her father’s statement?
A
B
C
D
Students’ ResponsesAll students translated the farmer statement into a system equations similar to:
Attempted to construct a 4th equation, e.g.,
200
400
800
B A
C B
D C
( 200) ( 400) ( 800)B C D A B C
Teacher’s action 5: Reflective public discussion:
• Why did our first approach to solving the problem fail?• The need to attend to the figure’s form:
versus
Objective 2:• To advance the way of thinking:
In representing a problem algebraically, all of the problem constraints must be represented.
A
B
C
D
A
CB
D
Some Essential Algebraic Ways of Thinking
7. Definitional reasoning• Definitions in terms of students’ (immediate)
experience
Some Essential Algebraic Ways of Thinking
8. Relating physical reality to algebraic reality to geometric reality
• GPA way of thinking
Some Essential Algebraic Ways of Thinking
9. Geometric thinking
The loss of geometry