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Recursion

Jeff Parker, Merrimack College

CS2 @ Dennison, June 2008

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

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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

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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…

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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.

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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

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ImagesImages 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

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Dan Gries' Fractal MakerA system that lets students play with different fractal images

Can select the depth of recursive calls

Has a nice library of exampleshttp://www.dangries.com/Flash/FractalMaker.html

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Dolls

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Images

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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.

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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?

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Other Motivating Examples

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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

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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

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Print a Stringvoid print(string str){

if (str.length() > 0){

char ch = str.at(0);string tail = str.substr(1, str.length());print(tail);cout << ch;

}}

[Ford] Call on students to act out rolesEach student has a different copy of this page

I think it is important to make tail explicit, rather than a function callif (str.length() > 0) { ...

print(str.substr(1, str.length()));

print("STOP");

str

ch

You can observe a lot just by watching - Yogi Berra

tail

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Reverse a StringTo focus on passive flow, assemble workIn 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;

}}

reverse("STOP");

str

ch

tail

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Teaching Models

[Wu, Dale, and Bethel] identify 5 teaching modelsThree 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 InductionStructure template – Sample solutions and a template that they can fill in

They believe that different students will prefer different approaches

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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

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Recursion in TextCS1 Students were given problem to reverse the elements of DNS name

They had struggled to write an iterative solution

w w w . d e n i s o n . e d u

w w w..e d u d e n i s o n

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Reverse a string w/ three words

w w w . d e n i s o n . e d u

string reverse3word(string str){

string::size_type pos = str.find(‘.’);string word = str.substr(pos+1, 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

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Reverse a stringstring reverse3word(string str){

string::size_type pos = str.find(‘.’);if (string::npos == pos)

return str;else {

string word = str.substr(pos+1, str.length();return reverse2word(word) + “.” + str.substr(0, pos);

}}string reverse2word(string str){

string::size_type pos = str.find(‘.’);if (string::npos == pos)

return str;else {

string word = str.substr(pos+1, str.length();return reverse1word(word) + “.” + str.substr(0, pos);

}}string reverse1word(string str){

return str;}

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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

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EROSI

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;

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

Tracing Calls: FillingConsider the task of filling a region

Call Tracking

Call trees are the gold standard [Kruse]Creating them requires solid understanding

// Fill Boolean array image at point (row, col)void fill(bool image[MAX][MAX], int row, int col){ // Is the fill point within the graph? if ((row < 0) || (row >= MAX) || (col < 0) || (col >= MAX)) return;

// Is the graph empty at the fill point? if (image[row][col]) return;

// Turn on the point image[row][col] = true;

// Try to fill in the surrounding area fill(image, row-1, col, level + 1);fill(image, row+1, col, level + 1);fill(image, row, col-1, level + 1);fill(image, row, col+1, level + 1);

}

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Mechanical Call Tracking

Simplest mechanical aid is to print info as we enter functionAdding indenting to show stack depth [Haynes]

void fill(bool image[MAX][MAX], int row, int col, int level){

cout << indent(level) << "fill(" << row << ", " << col << ")\n";...// Fill in the surrounding area fill(image, row-1, col, level + 1);fill(image, row+1, col, level + 1);fill(image, row, col-1, level + 1);fill(image, row, col+1, level + 1);

}

fill(4, 4)

fill(3, 4)

fill(2, 4)

fill(1, 4)

fill(3, 4)

fill(2, 3)

fill(2, 5)

fill(1, 5)

fill(3, 5)

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Mechanical Call TrackingHere we track which call (first, second, third…) filled each cell

void fill(bool image[MAX][MAX], int row, int col, int & cnt, int count[MAX][MAX]){

// Is the fill point within the graph?if ((row < 0) || (row >= MAX) || (col < 0) || (col >= MAX))

return;

// Is the graph empty at the fill point?if (image[row][col])

return;

// Turn on the pointimage[row][col] = true;count[row][col] = ++cnt;

// Try to fill in the surrounding area fill(image, row-1, col, cnt, count);fill(image, row+1, col, cnt, count);fill(image, row, col-1, cnt, count);fill(image, row, col+1, 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

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DramatizationsBen-Ari suggests that we dramatize recursionHe gives 3 examples. Each emphasizes the passive flow.

Open a present (Improvement on nested Matryoshka Dolls)The receiver thanks the presenter to show passive

flowBuild a chain of N links by adding a link on passive pathCount the nuts in a Chocolate Bar: assemble count

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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

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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