Impasse, Conflict Impasse, Conflict and Learning of CS Notions and Learning of CS Notions David Ginat Ginat David Tel- - Aviv University Aviv University Tel 32 slides 32 slides
Learning Learning Rote Learning Rote Learning Learning with understanding Learning with understanding Procedural knowledge Procedural knowledge Conceptual knowledge Conceptual knowledge
Two Examples Two Examples Average: Average: � How How to compute it to compute it � � What What does it mean does it mean � Iterative Computation: Iterative Computation: � How How is it executed is it executed � � What What are its characteristics are its characteristics �
Average Average � Compute the � Compute the avg avg of N of N nums nums � Given N � Given N- - 1 1 nums nums and and avg avg find the N- - th th num num find the N � Given K � Given K nums nums and and avg avg offer N- - K K additional additional nums nums offer N � Characterize the � Characterize the avg avg in terms of the in terms of the nums larger and smaller than it larger and smaller than it nums
Iterative Computation Iterative Computation � Construct a loop to compute � Construct a loop to compute … … � Given the following loop, offer: � Given the following loop, offer: - input(s input(s) that yields no iterations ) that yields no iterations - - input that yields K iterations input that yields K iterations - - input that yields infinite iterations input that yields infinite iterations - - a general relationship ( a general relationship ( e.g. ) - invariant ) e.g. invariant between its variables between its variables
Notion Utilization Notion Utilization Different types of tasks: Different types of tasks: Explicit reference to the notion Explicit reference to the notion No explicit reference No explicit reference but the notion is “ “ called for called for” ” but the notion is No explicit reference No explicit reference hidden relevance of the notion hidden relevance of the notion
Notions of this Talk Notions of this Talk Rigor Rigor in the design of argumentation in the design of argumentation �� Recursion Induction �� Recursion Induction in the design of an algorithm in the design of an algorithm
Board Staining Board Staining � � A board of N× ×N squares, N N squares, N- - 1 are stained. 1 are stained. A board of N A square with at least 2 stained neighbors A square with at least 2 stained neighbors becomes stained. Is there an initial becomes stained. Is there an initial staining that yields a stained board? staining that yields a stained board?
Board Staining Board Staining � � Eventually, only part of this board will Eventually, only part of this board will be stained be stained
Student (Teacher) Tendencies Student (Teacher) Tendencies “ Maximal initial structures, for which Maximal initial structures, for which … … no no “ other structure may stain more … … ” other structure may stain more ” Seems true, but how do you prove that? Seems true, but how do you prove that?
Student Tendencies Student Tendencies Try to prove by induction that “ “ there will there will Try to prove by induction that always be an unstained column and row” ” always be an unstained column and row Seems true, but how to apply the induction? Seems true, but how to apply the induction?
Student Tendencies Student Tendencies - Yield sound observations, but not patterns Yield sound observations, but not patterns - on which to capitalize on which to capitalize - Follow a single train of thought Follow a single train of thought - - Do not view a proof construction as Do not view a proof construction as - problem solving problem solving � Fixation, conflict � affective reaction � Fixation, conflict � affective reaction � Cognitive tension between the clear � Cognitive tension between the clear observations and the inability to convince observations and the inability to convince
Change the Point of View Change the Point of View � � Sole area examination yields no clue Sole area examination yields no clue The circumference circumference may also be relevant may also be relevant The
Invariant Property Invariant Property � � The number of stained circumference The number of stained circumference � Invariant sides does not increase! � Invariant sides does not increase!
Goal Cannot be Attained Goal Cannot be Attained � � Initially at most 4× ×(N (N- - 1) stained circum 1) stained circum - - sides sides Initially at most 4 At the end they need to be 4× ×N. Impossible! N. Impossible! At the end they need to be 4
Learning Learning Role of rigor: a rigorous pattern Role of rigor: a rigorous pattern yields convincing argumentation yields convincing argumentation Invariance property, and its link to Invariance property, and its link to the initial state and the final state the initial state and the final state Relevance of attempting various Relevance of attempting various points of view, not only the initial one points of view, not only the initial one
Learning by Conflict in Math Learning by Conflict in Math Infinity (e.g., Infinity (e.g., Sierpinska Sierpinska, 1987) , 1987) �� { 2, 4, 6, } �� { 1, 2, 3, … … } { 2, 4, 6, … …} } { 1, 2, 3, �� { (1,1), (1,2) } �� { 1, 2, 3, … … } { (1,1), (1,2) … … (2,1) (2,1) … …} } { 1, 2, 3, Epistemological obstacle (threshold concept?) Epistemological obstacle (threshold concept?) Proof elements (e.g., Proof elements (e.g., Movshovitz Movshovitz 1990) 1990) Sqrt of 2 is irrational of 2 is irrational Sqrt � Sqrt � Sqrt of 4 is irrational (deliberate errors) of 4 is irrational (deliberate errors)
Binary Sequence Binary Sequence w(1)= 0 w(2)= 001 w(1)= 0 w(2)= 001 w(i+ 1) is obtained from w(i w(i) by ) by w(i+ 1) is obtained from replacing 0 0 by 001 by 001 and and 1 by 0 1 by 0 replacing � w(3)= 0010010 � w(3)= 0010010 The value of the N- - th th bit in the first bit in the first The value of the N long- - enough word? enough word? long
Solution Attempts Solution Attempts � 001, 1 � 0 The rules: 0 � 001, 1 � 0 The rules: 0 w(1)= 0, w(2)= 001, w(3)= 0010010 w(1)= 0, w(2)= 001, w(3)= 0010010 � w(4)= 00100100010010001 � w(4)= 00100100010010001 Exponential growth Exponential growth Solution approaches: Solution approaches: Inductive simulation, 1’ ’s locations(?) s locations(?) Inductive simulation, 1
Student Tendencies Student Tendencies Seek variants of inductive progression Seek variants of inductive progression … but the required space is too large but the required space is too large … Seek patterns of the locations of 1’ ’s s Seek patterns of the locations of 1 … but no clear pattern but no clear pattern … � Fixation, Conflict � Fixation, Conflict � Cognitive tension, Epistemic curiosity � Cognitive tension, Epistemic curiosity w(4)= 00100100010010001 w(4)= 00100100010010001
Change the Point of View Change the Point of View � 001, 1 � 0 The rules: 0 � 001, 1 � 0 The rules: 0 w(1)= 0, w(2)= 001, w(3)= 001 001001 0010 0 w(1)= 0, w(2)= 001, w(3)= � w(4)= � w(4)= 0010010 00100100010010 0010010001 001 � w(i+ 1)= w(i)w(i)w(i � w(i+ 1)= w(i)w(i)w(i- - 1) 1) Recursive view, inductive validation inductive validation Recursive view, Base: √ √ Step: w(i w(i)= w(i )= w(i- - 1)w(i 1)w(i- - 1)w(i 1)w(i- - 2) 2) Base: Step:
Capitalize on the New Pattern Capitalize on the New Pattern w(1)= 0, w(2)= 001, w(3)= 0010010 w(1)= 0, w(2)= 001, w(3)= 0010010 w(4)= 0010010 00100100010010 0010010001 001 w(4)= w(i+ 1)= w(i)w(i)w(i- - 1) 1) w(i+ 1)= w(i)w(i)w(i � length(i+ 1)= 2 � length(i+ 1)= 2× ×length(i)+ length(i length(i)+ length(i- - 1) 1) The length grows exponentially, The length grows exponentially, � Keep a table of the word lengths � Keep a table of the word lengths
Compute Recursively Compute Recursively w(1)= 0, w(2)= 001, w(3)= 0010010 w(1)= 0, w(2)= 001, w(3)= 0010010 w(4)= 0010010 00100100010010 0010010001 001 w(4)= L(2)= 3, L(3)= 7, L(4)= 17, L(5)= 41 L(2)= 3, L(3)= 7, L(4)= 17, L(5)= 41 w(i+ 1)= w(i)w(i)w(i- - 1) 1) w(i+ 1)= w(i)w(i)w(i � 17< � w(5) bit 20? � < 41 � 17< 20 20< 41 w(5) bit 20? w(3) � � bit 3 in w(4) w(5)= w(4) w(4) w(4) w(3) bit 3 in w(4) w(5)= w(4) � bit 3 in w(3) w(3)w(2) � w(4)= w(3) w(3) w(3)w(2) bit 3 in w(3) w(4)=
Learning Learning �� Recursion Induction �� Recursion Induction Opposite directions Opposite directions But very close, incremental reasoning But very close, incremental reasoning Shown separately in CS studies Shown separately in CS studies Induction in iteration and proofs Induction in iteration and proofs Recursion in reverse computations and Recursion in reverse computations and data structures data structures
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