The Power of Recursion and Induction

In this brief we will focus on three domains to elaborate on the question of why teach recursion and induction: Within mathematics, in modeling, and in technology

1. Within mathematics a) Concepts and processes already described in a recursive or inductive

way. Recursion is everywhere in mathematics. For example, the set of natural numbers is defined recursively. Some functions, such as the Ackermann function are also defined recursively. Recursion appears also in natural phenomena. For example, the Fibonacci numbers: F(n) = F(n − 1) + F(n − 2) and F(1) = 1 and F(2) = 1 appear in several biological settings. Recursion appears also in the school curriculum. It is often used in relation to teaching the topics of Mathematical Induction and Sequences. In some countries recursion has a substantial presence in the school curriculum; in others it is marginal. Contemporary technological tools make it easy to use recursion. Moreover, they enable the recursion process to become visible.

What is the area of the following shape (called the Koch snowflake) (r is the diameter of the circle)? What is its perimeter?

This is a difficult problem to solve.

The following is a much easier problem. What is the area of this equilateral triangle (s being the length of its side)? What is its perimeter?

How can solving the simple triangle problem help in solving the complex one?

If you remove the middle one-third of each side of the equilateral triangle and replace it with two segments congruent to the segment you removed you will get the shape below. Can you find its area and perimeter?

If you continue this inductive process you will produce the Koch snowflake in the complex problem after a final number of steps. It is clear now how to calculate its area and perimeter.

Recursion is a useful method for solving certain complex problems that we don’t know how to solve. It is useful in cases when we can transform a complex problem using successive steps to a simpler problem, which we know how to solve. Together with induction it allows us to move from a complex to a simple problem and then back to solving the complex problem.

There are several reasons for explicit uses of recursive thinking and mathematical induction in the mathematics curriculum. Among them is the fact that many of the topics we obtained from the mathematics curriculum lend themselves to recursive thinking. So, as a context, recursive thinking may be used to show coherence in the development of these topics. For example, sequences, series, and tessellations can be developed using recursive thinking to show how subsequent values are obtained from previous values. Also the financial markets, which are huge and impact on many lives, use some mathematics to make sense of these markets. These aspects of mathematics that include topics like annuity, mortgage, present value, future value, all yield themselves to recursive thinking. Consequently, these topics may be taught meaningfully by using recursive thinking processes. It is worth noting that relationships derived from how to generate subsequent values from previous values can be verified and proved

by the process of mathematical induction to ensure that the relationships give reliable subsequent values, once the initial values are known.

b) Concepts and processes from other areas of mathematics, where recursion and induction can play an important role.

2 In modeling Per. Dealing with recursion enhances focus on modelling and hence relational thinking. Teaching recursion often takes the form of pupils having to find a recursive formula for a given problem e.g. number of diagonals in n-polygons. Used/taught this way, the process is actually a process of modelling the situation. The modelling process involves in general several stages, see (“Mathematical modelling and application”, W.Blum et al), but in particular one of the stages, “mathematization”, is often hard to do. This is the process where the students introduce the language of mathematics in order to solve the problem. There are two obvious benefits: one is that the hard job of mathematization is being emphasised and practiced, and the other is an aspect of mathematization: Relational thinking. In trying to identify and describe the main features of the exploited pattern focus is drawn to a deep understanding of what is going on and how it relates to the cognitional scheme already present in the students mind.

In technology:Within the technology itself Noi

Using technology Technology (the case of spreadsheets) Per:

When using a spreadsheet recursive processes will often occur. In the spreadsheet illustrated here, there are three recursive functions. We have, in column:A: fn=fn-1+1 B: fn=fn-1+300C: fn=fn-1·1.03+300

Real life application:A: yearsB: total amount, without interestC: total amount, with 3% interest per year.

The Fibonacci-sequence fn+2=fn+1+fn can easily be unfolded in a spreadsheet. Note that, by definition:A1=1A2=1

Another example of recursive functions in spreadsheets:

Consider the population number in year n (Pn) as a function of the population number in year n-1 (Pn-1), plus the number of newborns (the rate b·Pn-1), minus the number of deaths (using the rate d· Pn-1) plus the net immigration (in absolute numbers). Analytically, this would be a rather complicated function, but described as a recursive process, it is straight forward from the model above:

This can be described by: Pn=Pn-1+b·Pn-1-d·Pn-1+I = Pn-1·(1+b-d)+I

While the production of the Fibonacci numbers primarily will take place in a school context, the other examples could unfold in an everyday or professional context. There is very little doubt that the spreadsheet formulae that produces the numbers in the tables deals with recursivity, but an interesting question is, if the Spreadsheet users competence regarding these matters would be enhanced through training explicitly in the concept and notation of recursion. If the answer is yes, and it is agreed that the use of spreadsheet is becoming a common thing all over the world, this offers a strong argument for the teaching of recursive functions.’

Conclusion Ruhama

Appendix: Recursion in school curriculum Mi yeong#1From the square, First remove 1/4 square and you repeat each step like figure.Find the area of painted square. What’s nth term?

Step 0 1 2 3 … n …Number of colored squares

1 3 9 27 … 3n

Colored area 1 3/4 9/16 27/64 … 3n/4n

We can find without counting.It seems too complex, but if we use recursion, the problem is simpler.

As seen before, the number of colored squares is 1, 3, 9, 27, 81, ...The recursive formula is: an+1 =an×3

Sure this problem seems to be trivial.

But the problem of finding the colored area as a fraction of the original square is more complex.We think together one square’s area and square’s number. So this problem is why we use recursion.

1, 3/4, 9/16, 27/64, … This is an+1 =an×3/4

The use of recursion is simplified the complex problem.

#2. By using sticks, we can make hexagon-pole(6-pole).We can use the hexagon-pole(6-pole) to build more complex conglomerates as shown below. What is the number of sticks needed to build the 20th conglomerate? And what is the number of sticks needed to build the nth conglomerate?

If we count the number of sticks, Conglomerate 1 2 3 4 …Number of sticks 18 43 76 117

It seems too complex, but if we use recursion, solving the problem becomes easier .a2=a1+25a3=a2+33a4=a3+41…

25, 33, 41, … It is increasing by 8.So the general term is 25+8(n-1)=8n+17

Thus, the recursion formula is an+1=an +8n+17And the general term is an=4n2+13n+1

#3. You can think a right triangle that the length of side is a=5, b=4, c=3. And you can

make square using two sides except the hypotenuse of a

right- angled triangle like below. And we draw the similar right triangle like right triangle given at first. And repeat this process. Then what is the area of blue’s square? What is the area of green’s square?

the area of blue’s squarea1=32

a2=(5/9)2

…

Without special number, Let’s do like that. a1=c2

a2=(c2/a)2

hypotenuse of a

right angled triangle

a3=(c3/a2)2

…

an+1=an× (ca

)2

(Because the side’s recursive formula is bn+1=bn×ca

, the recursive formula of area of

rectangle is an+1=an× (ca

)2.

The area of square is square of length.)

We can see the technology by Java(turtle) and GSP using recursion.

<turtle microworld> http://www.javamath.com/class/

logo; system puzzle;

def phyt(a,n) {new;repeat 4 { pt; fd a ; rt 90 }; paint yellow; tile random;if (n>0) {fd a; rt 60;fd a*0.866 ;phyt(a*0.5,n-1);fd -a*0.866; rt -90;phyt(a*0.866,n-1);rt 30;fd -a;};}; def p(n) {CLS;tt; window 85;line size, 10;fd;tt 0,-35,90;phyt(20,n);but();};

def aphyt(a,n) {repeat 4 { fd a ; rt 90 };if (n>0) {fd a; rt 60;fd a*0.866 ; phyt(a*0.5,n-1);fd -a*0.866; rt -90;phyt(a*0.866,n-1);rt 30;fd -a;} else {};};def ap(n) {cls;tt green;tt 0,-45,90;aphyt(20,n);};def but() {button p(0) {0;-75,-47;black , 9 ; yellow,25};button p(1) {1;-50,-47;black , 9 ; yellow,25};button p(2) {2;-25,-47;black , 9 ; yellow,25};button p(3) {3; 0,-47;black , 9 ; yellow,25};button p(5) {5;25,-47;black , 9 ; yellow,25};button p(7) {7;50,-47;black , 9 ; yellow,25};button p(9) {9;75,-47;black , 9 ; yellow,25};button kkk { click the yellow button written by mouse ;-70,-58;blue,0; white,0}; };def pnamu() { point v_0,-65, 54; label v_0 { gif(phytatree.gif)} };p(1);

by Hanhyuk Cho, professor, Seoul university school

#4. Like figure, when regular hexagon is added and the points are increased, What is the number of nth points?

It’s difficult to find the general term. But we can find easier by using recursiona1=6a2=a1+2×4+1=6+9=15a3=a2+3×4+1=6+9+13=28a4=a3+4×4+1=6+9+13+17=45

…

an=an-1+n×4+1= an-2+4(n-1)+1+4n+1=…=a1+9+13+17+…+4n+1an=(n+1)(2n+1)

Proof by inductiona1=(1+1)(2+1)=6. True.

Suppose an is true.an=6+9+13+17+…+4n+1=(n+1)(2n+1)Then, an+1=6+9+13+17+…+4n+1+4(n+1)+1 =2n2+7n+6 =(n+2)(2n+3) ={(n+1)+1}{2(n+1)+1}This is true.So this is correct by induction.

When we solve the difficult and complex problem, we can find the general term or any term by using the recursion. Recursion is available the thinking’s pattern that can move from large to small.