Introducing recursion Monga Nothing to fear but fear itself: introducing Why recursion in lower secondary schools recursion? Why recursion is challenging Our Violetta Lonati Dario Malchiodi methodology Mattia Monga Anna Morpurgo Algomotricity Algomotorial recursion Results Dept. of Computer Science Università degli Studi di Milano aladdin@di.unimi.it LaTiCE — Hong Kong, April 23, 2017 1
Why recursion? Introducing recursion Recursion is a fascinating and powerful problem solving Monga technique. Why recursion? Ubiquitous in computer science (for examples in grammars) Why recursion is challenging Our methodology Algomotricity Algomotorial recursion Results 2
The power of recursion Introducing recursion If you can see the recursive “nature” of a problem you have Monga solved it! Why recursion? Move from A to C , no bigger disk upon a smaller one. . . Why recursion is challenging Our methodology Algomotricity Algomotorial recursion Results hanoi ( 1 , A , C ) = move ( A , C ) hanoi ( n , A , C ) = hanoi ( n − 1 , A , B ); move ( A , C ); hanoi ( n − 1 , B , C ) 3
Why recursion is challenging Introducing recursion Recursion is often believed out of the reach for lower secondary Monga schools: Why It is abstract: a recursive solution assumes a “leap of faith” recursion? in the solution of smaller versions of the problem. As a Why recursion is result, the solution seems magic, since the required steps challenging Our are not explicit. methodology Algomotricity It is given for a family of problems, not a single instance. Algomotorial recursion It breaks the common sense (and classic) rule: no Results 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! 4
Algomotricity Our own “allosteric” (Giordan, 1996) evolution of Introducing recursion “Computer Science Unplugged” (Bell et al., 1998) Monga Algomotricity Why Pupils are exposed to an informatic concept/process recursion? by playful activities, which imply a mix of tangible Why recursion is and abstract object manipulations: they can challenging investigate it firsthand, make hypotheses that can Our methodology then be tested in a guided context during the Algomotricity activity, and eventually construct viable mental Algomotorial recursion models. Results 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. 5
Algomotricity with LEGO bricks Introducing recursion Monga Why recursion? Execution of a recursive Why recursion is algorithm to compute the length challenging of a word, represented by a tower Our methodology of LEGO bricks (a brick is a Algomotricity Algomotorial letter). recursion Results Students are given instructions to do the computation. 6
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. 7
Algomotricity with LEGO bricks Introducing recursion Monga Why recursion? 1 each pupil execute a task according the instructions given Why recursion is (‘blind’ delegation) challenging Our 2 pupils discover soon that instructions are always “the methodology same”, since any sub-tower of bricks is still a tower Algomotricity Algomotorial recursion (self-similarity) Results 8
Then with a software tool Introducing The recursive strategy is then summarized as blind delegation recursion to helpers able to do only similar tasks . A software tool Monga implements this with little “fairies” doing the recursive reverse of Why a string of characters (Little people metaphor, Haynes, 1995). recursion? Why First level: just a black box doing the reverse recursion is challenging Our methodology Algomotricity Algomotorial recursion Results 9
Then with a software tool Introducing The recursive strategy is then summarized as blind delegation recursion to helpers able to do only similar tasks . A software tool Monga implements this with little “fairies” doing the recursive reverse of Why a string of characters (Little people metaphor, Haynes, 1995). recursion? Why First level: just a black box doing the reverse recursion is challenging Second level: the part of the string given to each fairy Our helper is made visible. The string is getting smaller going methodology Algomotricity to the right, it get larger coming back to the left. Three Algomotorial recursion phases are highlighted: reduction (yellow), base (red), Results reconstruction (orange). 9
Then with a software tool (cont.) Introducing recursion Monga Third level: each delegation can be explored in detail Why recursion? Why recursion is challenging Our methodology Algomotricity Algomotorial recursion Results 10
The activity in a school Introducing 18 pupils (8 th graders), 2 sessions for a total of 3 hours recursion Monga At the end of the activity, the class was able (with conductor help) to sketch a recursive algorithm to Why recursion? compute 2 18 Why recursion is 9 open questions worksheet to analyze actual challenging understanding (“how does the tower change when the Our methodology hourglass is yellow?”, “look at his picture: which letters are Algomotricity written in the tower, soon after the action of the fairy?”) Algomotorial recursion Results 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. 11
Conclusions Introducing recursion Monga Why We believe recursion is indeed accessible to lower recursion? secondary school pupils, our preliminary observations seem Why recursion is to confirm it. challenging Delegation + self-similarity can be enough to show most of Our methodology the power of recursive solutions, even at an early age. Algomotricity Algomotorial recursion Informatics is somewhat a “concrete mathematics”, but it is Results sometimes not concrete enough for young pupils: allosteric/kynesthetic strategies can be important. 12
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