Recursion

1
Recursion

• Jeff Parker, Merrimack College
• CS2 _at_ Dennison, June 2008

In order to understand recursion you just need
to understand recursion.
2
Outline

• Recursion is important
• We use it to describe things (BNF)
• Many algorithms are best expressed recursively
• Our choice is not if we should teach it, but how
  to teach it
• Recursion is hard
• Our students don't understand it
• Our students fear it
• This talk hopes to provide
• Review of the literature
• More ideas than you can possibly use, and
• A framework to work within

3
How to teach it

• There are two theories on how to teach it
  Students should
• Have a strong model of how it works, or
• Have a template for how to use it, without
  worrying so much about how it works Long, Weide,
  Bucci
• These are not as contradictory as they seem
• Understanding how it works well enough to predict
  what a new example does is important
• Understanding how it works does not help us solve
  new problems
• We need both skills. But we need more

4
Framework for teaching

• Motivation
• As with any new notion, the student must have a
  reason to master the new idea.
• Need a problem they cannot solve without
  recursion
• Mental model of recursion
• Must be strong enough to allow student to test
  alternatives.
• Understanding of how to apply recursion
• The mental model helps us simulate action of
  function.
• Does not help us write the function in the first
  place.

5
Motivation

• We want examples that grab the students
• We don't have to explain in full detail
• - that comes later
• Look for examples in the following areas
• Images
• Language
• Text pyramids, reverse ASCII "graphics"
• Games and Puzzles
• Avoid fib() and factorial()
• Prefer power(base, exp)
• As well as problems that recursion can help us
  solve, we explore the richness of recursion

6
Images

• Images are excellent at capturing the idea
• Some of the images are recursive by nature
• Some are easy to describe recursively
• Introducing APIs needed to write examples is
  barrier
• If we are using these as motivation this isn't an
  issue
• Some images are distracting
• As much as we like them, they confuse our students

7
Dan Gries' Fractal Maker

• A system that lets students play with different
  fractal images
• Can select the depth of recursive calls
• Has a nice library of examples
• http//www.dangries.com/Flash/FractalMaker.html

8
Dolls
9
Images
10
Recursion in Language

• Noam Chomsky believes that the ability to
  recursively extend a sentence is hardwired.
• This is the man all tattered and torn,
• That kissed the maiden all forlorn,
• That milked the cow with the crumpled horn,
• That tossed the dog, That worried the cat,
• That killed the rat, That ate the malt
• That lay in the house that Jack built.
• Khad Gadya
• One little goat that Father bought for two zuzim.
  
• Daniel Everett has proposed the Pirahã language
  as a counter-example.

11
Recursion in Language

• An interesting exercise in inductive reasoning is
  to discuss the statement
• A King is a son of a King
• Is this true? Why or why not?
• George I ascended after James II was deposed.
• Does this prove that Prince Charles can never
  take the throne?
• If we assume it is true, what else can we say
  about Kings?

12
Other Motivating Examples
13
Student Mental Models

• Kahney analyzed student mental models
• Looping model Recursion as a form of iteration
• Step Model just follow the program one step at
  a time
• Return Value Model Assumes each call returns at
  once
• Models in which Return returns to the main
• Magic Model No Clue
• Copies Model Distinct invocations of a routine
  (the only correct model)
• Each Slugo is different
• Since then, other models have been suggested.
• Part of our task is to render other models
  untenable

14
Copies Model

• George argues that to teach Copies model, we
  must
• Show active flow of control (from caller to
  callee)
• Show passive flow of control (callee returns to
  caller)
• Understand the base case
• We can explain all this with linear recursion
  (single call)
• Pure Tail Recursion does not have observable
  passive flow
• Embedded Recursion may

15
Print a String
You can observe a lot just by watching - Yogi
Berra
str

• void print(string str)
• if (str.length() gt 0)
• char ch str.at(0)
• string tail str.substr(1, str.length())
• print(tail)
• cout ltlt ch
• Ford Call on students to act out roles
• Each student has a different copy of this page
• I think it is important to make tail explicit,
  rather than a function call
• if (str.length() gt 0) ...
• print(str.substr(1, str.length()))

ch
tail
print("STOP")
16
Reverse a String
str

• To focus on passive flow, assemble work
• In this function, we build a string
• string reverse(string str)
• if (str.length() 0)
• return str
• else
• char ch str.at(0)
• string tail str.substr(1, str.length())
• return tail ch

ch
tail
reverse("STOP")
17
Teaching Models

• Wu, Dale, and Bethel identify 5 teaching models
• Three are concrete, two abstract
• Russian Dolls - smaller versions of the original
  (recursive call) and a doll that you cannot open
  (base case)
• Process Tracing Use call tree Kruse or
  activation trace Haynes
• Stack Simulation Show the underlying computer
  architecture used
• Mathematical Induction Mathematical examples
  and Proof by Induction
• Structure template Sample solutions and a
  template that they can fill in
• They believe that different students will prefer
  different approaches

18
Writing Recursive Routines

• The third leg of the stool is to write recursive
  routines
• When we present a recursive definition of the
  Factorial function, we have skipped the step that
  students find most difficult
• When we teach functions, we teach students to
  understand what the client needs, and to write a
  provider to implement it.
• Recursive routines are both client and provider
• Long, Weide, Bucci Apply this to your problem
  of choice

19
Recursion in Text

• CS1 Students were given problem to reverse the
  elements of DNS name
• They had struggled to write an iterative solution

w
w
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d
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o
n
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d
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w
w
w
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.
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n
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Reverse a string w/ three words

• string reverse3word(string str)
• stringsize_type pos str.find(.)
• string word str.substr(pos1, str.length()
• return reverse2word(word) . str.substr(0,
  pos)
• Of course, this does not work if there are no
  dots.
• Bullet-proof this with a check that pos is not
  negative

w
w
w
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d
e
n
i
s
o
n
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e
d
u
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Reverse a string

• string reverse3word(string str)
• stringsize_type pos str.find(.)
• if (stringnpos pos)
• return str
• else
• string word str.substr(pos1, str.length()
• return reverse2word(word) . str.substr(0,
  pos)
• string reverse2word(string str)
• stringsize_type pos str.find(.)
• if (stringnpos pos)
• return str
• else
• string word str.substr(pos1, str.length()
• return reverse1word(word) . str.substr(0,
  pos)

22
Visualizing the Call Stack
We have two copies of Pow3 on the stack This one
has a value of n 6 The arrow in bottom panel
shows where the program is running
The debugger shows the stack, for those that
understand it
23
EROSI

• While the debugger has all the information, we do
  not see the two calls to power as distinct
  objects
• George proposed the EROSI (Explicit Representer
  of Subprogram Invocations) system that shows each
  invocation as a distinct object

int main() p power(2, 13)
power(2, 13) temp power(2,
13/2) return temp
power(2, 6) temp power(2, 6/2)
return temp
24
Tracing Calls Filling

• Consider the task of filling a region

25
Call Tracking
25

• Call trees are the gold standard Kruse
• Creating them requires solid understanding
• // Fill Boolean array image at point (row, col)
• void fill(bool imageMAXMAX, int row, int col)
• // Is the fill point within the graph?
  
• if ((row lt 0) (row gt MAX) (col lt 0)
  (col gt MAX))
• return
• // Is the graph empty at the fill point?
  
• if (imagerowcol)
• return
• // Turn on the point
• imagerowcol true
• // Try to fill in the surrounding area
• fill(image, row-1, col, level 1)

26
Mechanical Call Tracking
26

• Simplest mechanical aid is to print info as we
  enter function
• Adding indenting to show stack depth Haynes
• void fill(bool imageMAXMAX, int row, int col,
  int level)
• cout ltlt indent(level) ltlt "fill(" ltlt row ltlt ", "
  ltlt col ltlt ")\n"
• ...
• // Fill in the surrounding area
• fill(image, row-1, col, level 1)
• fill(image, row1, col, level 1)
• fill(image, row, col-1, level 1)
• fill(image, row, col1, level 1)
• fill(4, 4)
• fill(3, 4)
• fill(2, 4)
• fill(1, 4)
• fill(3, 4)

27
Mechanical Call Tracking

• Here we track which call (first, second, third)
  filled each cell
• void fill(bool imageMAXMAX, int row, int col,
  int cnt, int countMAXMAX)
• // Is the fill point within the graph?
• if ((row lt 0) (row gt MAX) (col lt 0)
  (col gt MAX))
• return
• // Is the graph empty at the fill point?
• if (imagerowcol)
• return
• // Turn on the point
• imagerowcol true
• countrowcol cnt
• // Try to fill in the surrounding area
• fill(image, row-1, col, cnt, count)
• fill(image, row1, col, cnt, count)

0 0 0 0 0 0 0 0 0 0 0 0 0 X X X
X 0 0 0 0 0 X X 3 4 X X 0 0 0 X
X 15 2 5 6 13 X 0 0 X 26 16 1 X 7 12
X 0 0 X 25 17 23 X 8 11 X 0 0 0 X 18
22 X 9 10 X 0 0 X 20 19 21 24 X 14 X 0
0 0 X X X X X X X 0 0 0 0 0 0 0
0 0 0 0
28
Dramatizations

• Ben-Ari suggests that we dramatize recursion
• He gives 3 examples. Each emphasizes the passive
  flow.
• Open a present (Improvement on nested Matryoshka
  Dolls)
• The receiver thanks the presenter to show passive
  flow
• Build a chain of N links by adding a link on
  passive path
• Count the nuts in a Chocolate Bar assemble count

29
Summary

• Motivation
• Give students a reason to master a new idea.
• Mental model of the process
• Provide a mental model that they can use
• Reinforce with call tracing, call trees, etc.
• Understanding of how to apply recursion
• Use functional decomposition to break problem
  down
• Templates may help should not be straight-jacket

30
Selected Sources

• Ford A framework for teaching Recursion, SIGCSE
  Bulletin, 1982.
• Kahney Modelling novice programmer behaviour.
  In New horizons in educational computing,1984.
• Haynes Explaining Recursion to the
  Unsophisticated, SIGCSE Bulletin, September 1995
• Ben-Ari Recursion From Drama to Program,
  Journal of Computer Science Education, 11(3),
  1997
• Wu, Dale, Bethel Conceptual Models and
  Cognitive Learning Styles. SIGCSE 98
• Long, Weide, Bucci ClientView First An Exodus
  From Implementation-Biased Teaching, SIGCSE 1999
• George EROSI Visualizing Recursion and
  Discovering New Errors, SIGCSE 2000