Junior High Students Reasoning on Number Patterns

1
Junior High Students Reasoning on Number Patterns

• Fou-Lai Lin, Kai-Lin Yang, linfl team
• Department of Mathematics,
• National Taiwan Normal University

2
Outline

• Purpose
• Relevant Perspective
• Method
• Result and Discussion
• Reasoning of Number Patterns
• Reasoning of Mathematical Statements about Number
  Patterns
• Conclusion

3
Purpose of Study

• To investigate junior high students reasoning of
  number patterns
• To investigate junior high students reasoning of
  mathematical statements about number patterns

4
Relevant Perspective (Reasoning of Number
Patterns)

• Students Strategies of Reasoning of Number
  Patterns (Bishop, 2000)
• - Concrete
• - Proportional
• - Recursive
• - Functional
• Reasoning of Number Patterns
• - Task-comprehension (Concrete)
• - Generalization (Recursive, Functional)
• - Symbolization (Beyond Functional)
• - Checking
• (Proportional, Recursive, Functional,
  Beyond Functional)

5
Relevant Perspective(Reasoning of Mathematical
Statements about Number Patterns)

• Three Phases (Healy Hoyles, 2000)
• - Conception
• Own Approach
• Best Mark
• - Validation
• Rightness
• True
• Sometimes True
• - Construction
• Specializing a given statement
• Generalizing an argument
• Prove or Disprove

6
Method

• Developing three Questionnaires
• Sources
• England (C. Hoyles)
• Justifying and Proving in School Math
• Longitudinal Proof Project
• Taiwan (F.L. Lin)
• Junior High Students Mathematical Argumentation
• Framework
• Pilot Study
• Developing Coding System
• Modifying Coding and Items
• Conjecturing
• National Survey

7
Result(Reasoning of Number Patterns)

• G7 G8 Performance of Reasoning of Number
  Patterns
• The correct response frequency () of G7/G8
• (30.4/14.3) Incorrectly Generalizing with
  Proportional Strategies
• (15.0/21.3) Correctly Checking with Simple
  Property of Number Patterns

8
Result(Reasoning of Number Patterns)

• Developmental Hierarchy
• ( Linear and Quadratic Patterns)

Generalization
Symbolization
Task- Comprehending
9
Result(Reasoning of Number Patterns)

• Developmental Hierarchy in Linear Patterns

Generalization
Symbolization
Task- Comprehending
Checking
10
Result(Reasoning of Number Patterns)

• Developmental Hierarchy in Quadratic Patterns

Generalization
Symbolization
Task- Comprehending
Checking
11
Discussion(Reasoning of Number Patterns)

• H.L.T. for Teaching and Learning

12
Result (Reasoning of Mathematical Statements
about Number Patterns)

• The arguments with symbolic modes were most
  popular for own approach and for best mark

13
Result (Reasoning of Mathematical Statements
about Number Patterns)

• The performance of constructing proof in Grade 9

14
Result (Reasoning of Mathematical Statements
about Number Patterns)

• Most of the 9th graders proved with empirical
  mode.

15
Result (Reasoning of Mathematical Statements
about Number Patterns)

• Easy to Know, Hard to Do (????) in Formal Proof

16
Result (Reasoning of Mathematical Statements
about Number Patterns)

• The arguments with counterexample were not most
  popular for own approach and for best mark

17
Result (Reasoning of Mathematical Statements
about Number Patterns)

• Most of the 9th graders disproved with
  counterexample

18
Result (Reasoning of Mathematical Statements
about Number Patterns)

• Hard to Know, Easy to Do(????) in Disproof with
  Counterexample

19
Result (Reasoning of Mathematical Statements
about Number Patterns)

• Validation of Correct Proof
• About one-third of 9th graders agree that
  correct proofs with symbolic or narrative mode
  only show that the statement is true for some
  cases.

20
Result (Reasoning of Mathematical Statements
about Number Patterns)

• Validation of Incomplete Proof
• About one-third of 9th graders disagree that
  incomplete proofs with empirical or symbolic mode
  only show that the statement is true for some
  cases.

21
Result (Reasoning of Mathematical Statements
about Number Patterns)

• The mean of Validity Scores is associated with
  proof - conception

22
Result (Reasoning of Mathematical Statements
about Number Patterns)

• The mean of validity scores increased with
  proof-construction scores

23
Discussion (Reasoning of Mathematical Statements
about Number Patterns)

• H.L.T. in Proof (Easy to Know, Hard to Do)

Conception
Construction
Validation
24
Discussion (Reasoning of Mathematical Statements
about Number Patterns)

• H.L.T. in Disproof (Hard to Know, Easy to Do)
• Confliction between spontaneous and scientific
  conception of refutation
• What approach is beyond a conventional view of
  knowing disproof with one counterexample

Construction
Conception
? ? ?
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Conclusion

• Reasoning of number patterns
• Linear and Quadratic
• Discussion for teaching experiments
• Two Kinds of Checking Activity

Generalization
Symbolization
Comprehending
Checking (Linear)
Checking (Quadratic)
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Conclusion

• Reasoning of mathematical statements about number
  patterns
• Formal Proof
• Easy to Know(Conception) , Hard to
  Do(Construction)
• Disproof with Counterexample
• Hard to Know(Conception) , Easy to
  Do(Construction)
• Discussion for teaching experiments in proof
• Validation Activity

27
Grade 8 - A2

• A2. Karen and Josie are looking at these first
  four patterns in a sequence of dot patterns
• (a) The number of dots in the 4th pattern
  (Task-Comprehension)
• (b) The number of dots in the 20th pattern
  (Generalize)
• (c) Write an expression for the number of dots in
  the nth pattern (Sym.)
• (d) Do 9999 dots fit into this pattern (Check)

28
Grade 9-A1

• A, B, C, D, E are trying to prove whether the
  following statement is true or false
• When you add any 2 even numbers, your answer is
  always even.

A (Formal proof) a and b are any whole number 2a
and 2b are any two numbers 2a2b2(ab) So its
true.
B (Empirical) 22 4, 24 6, 42 6, 44
8 So its true.
C (Narrative) When you add numbers with a
common factor. So its true.
D (Narrative) Even numbers end in 0, 2, 4, 6 or
8. So its true.
29
Grade 9-A1

• Conception
• Choose one which would be closest to what you
  would do if you were asked to answer this
  question
• Choose the one which your teacher would give the
  best mark
• Validation
• As answer
• Has a mistake in it
• Shows that the statement is true
• Only shows that the statement is true for some
  even numbers
• Construction
• When you add any 2 odd numbers, your answer is
  always even.

30
Grade 9-A1

• A, B, C, D, E are trying to prove whether the
  following statement is true or false
• When you add any 2 even numbers, your answer is
  always even.

F (Formal proof) a is an even number, and the
next even number is a2, a(a2) 2(a1) So its
true.
G (Formal proof) x is any odd number, and x1,
x-1 are even numbers, (x1)(x-1) 2x So its
true.
H (Formal proof) 2x is an even number, 2x 2x
4x 2 2x So its true.
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Framework of Item Analysis

• Reasoning of Number Patterns

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Framework of Item Analysis

• Reasoning of Mathematical Statements about Number
  Patterns

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A Multi-Dimensional Hypothetical Learning
Trajectory
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Result

• Task-Comprehending VS. Generalizing (Grade 8-A2)

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Result

• Generalizing VS. Symbolizing (Grade 8-A2)

36
Result

• Generalizing VS. Checking (Grade 8 - A1)

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Result

• Generalizing VS. Checking (Grade 8-A2)

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Result

• Checking VS. Symbolizing (Grade 8-A2)

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Checking 1

• Aim at Generalizing
• Activity
• Drawing Concrete Patterns
• Assessing whether a number of dots fit into this
  pattern
• Strategy
• Concrete
• Proportional
• Recursive
• Functional

40
Checking 2

• Aim at Symbolizing
• Activity
• Drawing a general item of patterns with pictorial
  representations
• Assessing whether given expressions can represent
  this pattern
• Making patterns according to given expressions
• Representing patterns with symbols
• Strategy
• Functional
• Beyond Functional

41
Validation

• Local Starting
• Formal Proof Conception
• Local Aim
• Constructing Formal Proof
• Activity
• Recognizing Representation of Number Patterns
• (One symbol 3(a1)?6(2x-5), Multiple symbols
  2a4b)
• Evaluating the generality of arguments with
  symbolic modes
• Reasoning Operator
• Inductive
• Deductive

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Sample Size
43
Evaluation of Written Protocol
44
Evaluation of Written Protocol
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Validity Score

• Based on students responses to the following 3
  questions about argument
• Has a mistake in it
• Shows that the statement is true
• Only shows that the statement is true for some
  even numbers

46
Grade 8 - A1

• A1.Lisa has some white rectangle tiles and some
  green square tiles. The white tiles are twice as
  ling as the green tiles but have the same width.
• He makes a row of white tiles, like this
• He then builds a bridge of green tiles over the
  white tiles, like this

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Result

• Proof-Conception Quadratic Structure

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Knowing Doing
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Relevant Perspective(Reasoning of Mathematical
Statements about Number Patterns)
Reasoning Operators
Information
New Information

• Analogue
• Visual Reasoning
• (Transformation)
• Deduction
• Induction