Understanding mathematical proof and the validity of proofs in elementary school

Učenci se v zadnjem triletju osnovne šole srečajo z utemeljitvami ali argumentacijami. Te so lahko neformalne in temeljijo na argumentih, ki imajo ali pa nimajo matematične veljave, ali pa tudi preprostejše formalne. Temeljna vloga matematičnega dokaza pri pouku je spodbujanje matematičnega razumevanja, saj je dokaz najbolj prepričljiv, ko vodi do razumevanja. Pomembna matematična aktivnost je tudi prepoznavanje veljavnosti dokaza, ki zajema premislek o tem, ali je domnevni dokaz matematično pravilen in ali vključuje ustrezno logično matematično sklepanje. Raziskave so pokazale, da učenci ne čutijo potrebe po dokazovanju, doživljajo postopek dokazovanja kot težak in imajo največ težav ravno pri nalogah, ki zahtevajo utemeljevanje in dokazovanje trditev. V teoretičnem delu magistrskega dela smo obravnavali pojem matematičnega dokaza ter opisali njegovo vlogo pri matematiki in pri učenju matematike. Obravnavali smo pomen in vključevanje dokaza pri pouku ter predstavili kognitivne procese pri dokazovanju z vidika učenčevega razvoja. Predstavili smo različne funkcije dokaza v matematiki in pri pouku matematike, kognitivne vidike dokazovanja ter težave učencev pri učenju dokazov in dokazovanju. Predstavili smo tudi dve klasifikaciji matematičnih dokazov: klasifikacijo po Tallu in klasifikacijo glede strogosti utemeljitve. V empiričnem delu magistrskega dela smo uporabili kvantitativni in kvalitativni pristop za ugotavljanje izbranih vidikov dokazovanja v osnovnošolski matematiki. Zaradi vpliva učbenika kot vodila znanja na učence smo po dveh izbranih klasifikacijah analizirali zastopanost različnih vrst utemeljitev v najbolj razširjenih učbenikih za matematiko v zadnjem triletju osnovne šole. Analiza učbenikov »Skrivnosti števil in oblik« je pokazala, da je po klasifikaciji po Tallu v učbenike vključenih največ vizualnih dokazov, torej utemeljitev, ki imajo slikovno in verbalno podporo. Po klasifikaciji glede strogosti utemeljitve pa je v učbenike vključenih največ generičnih dokazov. Najmanj vključenih utemeljitev po klasifikaciji po Tallu je enaktivnih dokazov, po klasifikaciji glede strogosti utemeljitve pa utemeljitev z avtoriteto. Glede na vključenost utemeljitev pri različnih temah je največ utemeljitev vključenih pri temi geometrija in merjenje. Vzorec raziskave je predstavljalo 49 devetošolcev iz dveh različnih slovenskih osnovnih šol. Od tega je bilo v vzorcu 27 učno zmožnejših in 22 učno šibkejših učencev. S preizkusom znanja smo raziskali, ali učenci osnovnih šol sprejmejo in prepoznajo veljavnost utemeljitve matematičnega dejstva, torej nas je zanimalo, ali učenci razlikujejo med formalnim dokazom in neformalnimi utemeljitvami. Prav tako smo raziskali, ali razumejo različne vrste utemeljitev po dveh izbranih klasifikacijah. Podatki, zbrani s preizkusom znanja, so pokazali, da imajo učenci težave pri sprejemanju in prepoznavanju veljavnosti utemeljitev, saj je velik delež učencev kot veljavne utemeljitve prepoznalo večino vrst utemeljitev na preizkusu znanja. Kot najbolj razumljive vrste utemeljitev je največji delež učencev po klasifikaciji po Tallu prepoznal enaktivne utemeljitve, po klasifikaciji glede strogosti utemeljitve pa utemeljitev z avtoriteto in utemeljitev s sliko. Kot najmanj razumljivi vrsti utemeljitev pa je največji delež učencev po klasifikaciji po Tallu prepoznal manipulativni dokaz, po klasifikaciji glede strogosti utemeljitve pa formalni dokaz. Statistična analiza podatkov je pokazala, da obstajajo statistično pomembne razlike med učno zmožnejšimi in učno šibkejšimi učenci glede prepoznavanja ter sprejemanja veljavnosti in razumevanja različnih vrst utemeljitev glede na izbrani klasifikaciji. Z anketnim vprašalnikom smo raziskali stališča učencev o dokazih in dokazovanju pri pouku matematike. Malo več kot polovici učencev je težko podati utemeljitev neke matematične izjave, a hkrati menijo, da imajo dovolj znanja za razumevanje dokaza neke matematične izjave. Največji delež učencev se najmanj strinja s trditvijo, da običajno težko razumejo dokaz, ki ga učitelj predstavi pri pouku. Večji delež učencev ve, kaj je pri matematiki dokaz in kako je ta videti, ter zna presoditi, ali je nek dokaz pravilen ali ne. Učenci so izkazali želje po dokazih in dokazovanju pri matematiki. Več kot polovica učencev si želi, da bi se pri pouku predstavilo in opisalo več dokazov matematičnih izjav, pogovarjalo o resničnosti matematičnih izjav ter reševalo naloge, kjer se dokaže neko matematično izjavo. Zelo velik delež učencev si želi, da bi šolski učbenik vključeval več dokazov matematičnih izjav, in meni, da so dokazi zanimivi. Malo manj kot polovica učencev se strinja, da jim učitelj pri pouku pove, zakaj sploh dokazujemo matematične izjave, in meni, da če učitelj pri pouku matematično izjavo napiše na tablo, je trditev s tem že dokazana. Prav tako večji delež učencev meni, da so dokazi le za učence z boljšimi ocenami in da matematične izjave ni treba dokazati, če je zapisana v učbeniku. Velik delež učencev meni, da poznajo namen dokazovanja v matematiki, in se strinja, da je dokaz v matematiki pomemben, ker z njim zagotovimo resničnost izjave in ker nam razloži, zakaj neka trditev drži. S podatki, zbranimi s pomočjo anketnega vprašalnika, smo raziskali tudi, ali obstajajo statistično pomembne razlike med učno zmožnejšimi in učno šibkejšimi učenci glede stališč o dokazih in dokazovanju pri pouku matematike. Zbrani podatki so pokazali, da je večjemu deležu učno šibkejših učencev težje podati utemeljitev neke matematične izjave, da običajno težje razumejo dokaz, ki ga učitelj predstavi pri pouku, ter da se jim zdi, da nimajo dovolj znanja, da bi razumeli dokaz neke matematične izjave. Večji delež učno zmožnejših učencev se bolj od učno šibkejših učencev strinja, da vedo, kaj je pri matematiki dokaz in kako je ta videti, ter da običajno znajo presoditi, ali je nek dokaz pravilen ali ne. Večji delež učno zmožnejših učencev je pokazal večjo željo po tem, da bi šolski učbenik in pouk matematike vključevala več dokazov matematičnih izjav in nalog, kjer se dokaže neko matematično trditev. Večji delež učno šibkejših učencev se strinja, da če učitelj pri pouku neko trditev zapiše na tablo ali pa če je matematična izjava zapisana v učbeniku, te ni treba dokazati, saj je s tem že dokazana. Večji delež učno zmožnejših učencev se bolj od učno šibkejših učencev zaveda namena dokazovanja pri matematiki in pomembnosti dokaza, da z njim zagotovimo resničnost izjave in razložimo, zakaj neka trditev drži. Večji delež učno šibkejših učencev se bolj od učno zmožnejših učencev strinja, da dokazi niso pomembni za dobro znanje matematike ter da so dokazi le za učence z boljšimi ocenami. Tako učno zmožnejši kot učno šibkejši učenci so pokazali podobno stopnjo strinjanja s tem, da jim učitelj pri pouku pove, zakaj sploh dokazujemo matematične izjave.In the last three years of elementary school, pupils encounter substantiations or argumentations. These can be informal and based on arguments, which have or do not have mathematical validity, or even simpler formal ones. The basic role of mathematical proof in lessons is to encourage mathematical understanding, since a proof is most convincing, when it leads to understanding. An important mathematical activity is also the recognition of the validity of proofs, which includes the consideration, if an alleged proof is mathematically correct and if it includes a suitable logical mathematical reasoning. Research has shown that pupils do not feel the need for proving, they experience the proving procedure as hard and encounter most problems particularly with tasks, which require argumentations and proving. In the theoretical part of the master’s thesis, we discussed the concept of mathematical proof and described its role in mathematics and learning mathematics. We discussed the importance and incorporating proofs in lessons and presented the cognitive processes in proving from the perspective of the pupil’s development. We presented different functions of proof in mathematics and mathematics lessons, cognitive perspectives of proving and problems of pupils in learning proofs and proving. We also presented two classification of mathematical proofs: the classification according to Tall and the classification regarding the strictness of substantiation. In the empirical part of the master’s thesis, we used a quantitative and qualitative approach to identify the selected perspectives of proving in elementary school mathematics. Due to the influence of the textbook as a knowledge guide for pupils, we chose two classifications and used them to analyse different types of argumentations in the most widespread textbooks for mathematics in the last three years of elementary school. The analysis of the textbooks “Skrivnosti števil in oblik” has shown that according to the classification by Tall, the textbooks include mostly visual proofs, which means argumentations with figure and verbal support. According to the classification regarding the strictness of substantiation, in textbooks prevail generic proofs. The least common argumentations, according to the classification by Tall, are enactive proofs, and according to the classification regarding the strictness of substantiation, argumentations with authority. In terms of inclusion of argumentations in different topics, the most argumentations are included in the topic Geometry and Measuring. The research included a sample of 49 ninth graders from two different Slovenian elementary schools, out of which 27 were high achievers and 22 were low achievers. With a written examination, we analysed, if pupils accept and recognise the validity of an argumentation in a mathematical proof, meaning if pupils differentiate between a formal proof and informal argumentations. We also analysed, if they understand different types of argumentations according to the two chosen classifications. The data, obtained with the written examination, showed that pupils have problems to accept and recognise the validity of argumentations, since at the examination, a high proportion of pupils recognized the majority of types of argumentations as valid. According to the classification by Tall, the highest proportion of pupils recognised enactive argumentations as the easiest to understand, and argumentations with authority and argumentations with a figure according to the classification regarding the strictness of substantiation. As hardest to understand types, the highest proportion of pupils recognized the manipulative proof, according to the classification by Tall, and the formal proof, according to the classification regarding the strictness of substantiation. The statistical analysis of data showed that there are statistically significant differences between high and low achievers regarding the recognition and acceptance of validity and understanding of different types of argumentations in terms of the chosen classifications. We used a questionnaire to research the views of pupils on proofs and proving in mathematics lessons. A little more than half of the pupils find it hard to argument a certain mathematical statement, but at the same time they think they have enough knowledge to understand mathematical proofs. The highest proportion of pupils least agree with the statement that usually they find it hard to understand a proof presented by the teacher during lessons. A high proportion of pupils claim to know, what a proof in mathematics is and how it looks like, and know how to estimate, if a proof is correct or incorrect. The pupils expressed desires for learning proofs and proving in mathematics. More than a half of pupils would like that mathematical lessons would include more presentations, descriptions of proofs of mathematical statements, they would like to discuss about the correctness of mathematical statements and to solve proving tasks. A very high proportion of pupils want that the school textbook would include more proofs of mathematical statements and they think that proofs are interesting. A little less than a half of pupils agree that during lessons, the teacher tells them, why we bother to prove mathematical statements, and think that if the teacher writes a mathematical statement on the board during lessons, the statement is already proven. A high proportion of pupils also think that proofs are only for pupils with better grades and that, if a mathematical statement is written in the textbook, it does not require to be additionally proven. A high proportion of pupils think that they know the purpose of proving in mathematics and agree that a proof in mathematics is important, since it assures validity of the statement and explains, why a certain statement is correct. With the data, obtained with the questionnaire, we also analysed, if there are statistically important differences between high and low achievers regarding views on proofs and proving in mathematics lessons. The obtained data showed that most low achievers find it hard to argument a mathematical statement, usually find it difficult to understand a proof presented by the teacher during lessons, and find that they do not have enough knowledge to understand a proof of a mathematical statement. A significantly higher proportion of high achievers than low achievers agree that they know, what a proof in mathematics is and how it looks like, and that they usually can estimate, if a certain proof is correct or incorrect. Also, a significantly higher proportion of high achievers than low achievers expressed a desire for the school textbook and mathematics lessons to include more proofs of mathematical statements and more proving tasks. On the other hand, a significantly higher proportion of low achievers than high achievers agree that if a teacher during mathematics lessons writes a statement on the board or if the mathematical statement is written in the textbook, it does not require to be additionally proven and is therefore already proven. A higher proportion of high achievers than low achievers is aware of the purpose of proving in mathematics and the importance of a proof to assure the correctness of a claim. A higher proportion of low achievers than high achievers agree that proofs are not important for a good mathematics knowledge and that proofs are only for pupils with good grades. Both group of students expressed a similar level of agreement with the fact that during lessons, the teacher tells them, why we prove mathematical statements