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Introducing recursion Monga Why recursion? Why recursion is challenging Our methodology

Algomotricity Algomotorial recursion

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Nothing to fear but fear itself: introducing recursion in lower secondary schools

Violetta Lonati Dario Malchiodi Mattia Monga Anna Morpurgo

• Dept. of Computer Science

Università degli Studi di Milano aladdin@di.unimi.it

LaTiCE — Hong Kong, April 23, 2017

Introducing recursion Monga Why recursion? Why recursion is challenging Our methodology

Algomotricity Algomotorial recursion

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Why recursion?

Recursion is a fascinating and powerful problem solving technique. Ubiquitous in computer science (for examples in grammars)

Introducing recursion Monga Why recursion? Why recursion is challenging Our methodology

Algomotricity Algomotorial recursion

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The power of recursion

If you can see the recursive “nature” of a problem you have solved it! Move from A to C, no bigger disk upon a smaller one. . . hanoi(1, A, C) = move(A, C) hanoi(n, A, C) = hanoi(n − 1, A, B); move(A, C); hanoi(n − 1, B, C)

Introducing recursion Monga Why recursion? Why recursion is challenging Our methodology

Algomotricity Algomotorial recursion

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Why recursion is challenging

Recursion is often believed out of the reach for lower secondary schools: It is abstract: a recursive solution assumes a “leap of faith” in the solution of smaller versions of the problem. As a result, the solution seems magic, since the required steps are not explicit. It is given for a family of problems, not a single instance. It breaks the common sense (and classic) rule: no definiendum in definiens (but the base case avoids circularity). Thus, our main goal was to convince young pupils that recursion can be indeed a valid and useful strategy!

Introducing recursion Monga Why recursion? Why recursion is challenging Our methodology

Algomotricity Algomotorial recursion

Results

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Algomotricity

Our own “allosteric” (Giordan, 1996) evolution of “Computer Science Unplugged” (Bell et al., 1998) Algomotricity Pupils are exposed to an informatic concept/process by playful activities, which imply a mix of tangible and abstract object manipulations: they can investigate it firsthand, make hypotheses that can then be tested in a guided context during the activity, and eventually construct viable mental models. Role of computers and apps The computer is never a starting point, but all activities end with a computer-based phase in which participants use specific software tools that we have developed.

Introducing recursion Monga Why recursion? Why recursion is challenging Our methodology

Algomotricity Algomotorial recursion

Results

6

Algomotricity with LEGO bricks

Execution of a recursive algorithm to compute the length

• f a word, represented by a tower
• f LEGO bricks (a brick is a

letter). Students are given instructions to do the computation.

When the mate on your right asks to establish how long a word is, follow these instructions: if the word has one letter only whisper 1 to your mate. else take off a letter from the word and put it in the trash; pass the rest of the word to the mate on your left; ask him/her to establish how long it is; wait for his/her answer; add 1 to his/her answer; whisper the result to the mate who had asked you.

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Introducing recursion Monga Why recursion? Why recursion is challenging Our methodology

Algomotricity Algomotorial recursion

Results

8

Algomotricity with LEGO bricks

1 each pupil execute a task according the instructions given

(‘blind’ delegation)

2 pupils discover soon that instructions are always “the

same”, since any sub-tower of bricks is still a tower (self-similarity)

Introducing recursion Monga Why recursion? Why recursion is challenging Our methodology

Algomotricity Algomotorial recursion

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9

Then with a software tool

The recursive strategy is then summarized as blind delegation to helpers able to do only similar tasks. A software tool implements this with little “fairies” doing the recursive reverse of a string of characters (Little people metaphor, Haynes, 1995). First level: just a black box doing the reverse

Introducing recursion Monga Why recursion? Why recursion is challenging Our methodology

Algomotricity Algomotorial recursion

Results

9

Then with a software tool

The recursive strategy is then summarized as blind delegation to helpers able to do only similar tasks. A software tool implements this with little “fairies” doing the recursive reverse of a string of characters (Little people metaphor, Haynes, 1995). First level: just a black box doing the reverse Second level: the part of the string given to each fairy helper is made visible. The string is getting smaller going to the right, it get larger coming back to the left. Three phases are highlighted: reduction (yellow), base (red), reconstruction (orange).

Introducing recursion Monga Why recursion? Why recursion is challenging Our methodology

Algomotricity Algomotorial recursion

Results

10

Then with a software tool (cont.)

Third level: each delegation can be explored in detail

Introducing recursion Monga Why recursion? Why recursion is challenging Our methodology

Algomotricity Algomotorial recursion

Results

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The activity in a school

18 pupils (8th graders), 2 sessions for a total of 3 hours At the end of the activity, the class was able (with conductor help) to sketch a recursive algorithm to compute 218 9 open questions worksheet to analyze actual understanding (“how does the tower change when the hourglass is yellow?”, “look at his picture: which letters are written in the tower, soon after the action of the fairy?”) the answers that asked for a precise prediction were almost always correct; when they asked to verbally describe some features of the algorithm were still basically correct but in some cases they appeared vague or incomplete, but by discussing them with pupils, they had no difficulties in understanding and admitting the missing parts.

Introducing recursion Monga Why recursion? Why recursion is challenging Our methodology

Algomotricity Algomotorial recursion

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Conclusions

We believe recursion is indeed accessible to lower secondary school pupils, our preliminary observations seem to confirm it. Delegation + self-similarity can be enough to show most of the power of recursive solutions, even at an early age. Informatics is somewhat a “concrete mathematics”, but it is sometimes not concrete enough for young pupils: allosteric/kynesthetic strategies can be important.