Acta Didactica Universitatis Comenianae Mathematics, Issue 6, 2006
PROBLEMS WITH USING CALCULATORS ON MATHEMATICAL LESSONS1 ZUZANA BRISUDOVÁ, MÁRIA SLAVÍČKOVÁ
Abstract. The article arose during our stay in Norwegian Bodø during The Socrates Comenius 2.1 Project as a reaction on an excessive use of calculators at Math lessons. The experiment was realized in two Norwegian classes and one Slovak class. Our goal was to show that an excessive use of calculators could cause the reduction of the ability of results estimate of simple mathematical tasks, respectively an exact calculation of mathematical tasks. Résumé. Cet article a été créé pendant notre séjour à Bodø, en Norvège, dans le cadre du projet Socrates Comenius 2.1 comme la réaction à une utilisation exagérée des calculatrices durant des cours des Mathématiques. Expérimentation a été réalisée dans deux classes norvégiennes et dans une classe slovaque. Notre but était de montrer que l'utilisation exagérée des calculatrices peut causer la diminution de la capacité de faire l'estimation des résultats des tâches simples de mathématiques, ainsi que des calculs exacts des tâches mathématiques. Zusammenfassung. Dieser Artikel entstand im Projekt Socrates Comenius 2.1 während unseres Aufenthalts im norwegischen Bodø, als Reaktion an ein übermäßiges Benützen von Taschenrechnern. Das Experiment wurde in zwei norwegischen und einer slowakischen Klasse realisiert. Wir haben uns den Versuch, dass ein übermäßiges Benützen von TaschenRechnern eine Senkung der Fähigkeit von Ergebnisschätzungen in einfachen mathematischen Beispielen, bzw. des korrekten genauen Rechnens mathematischer Beispiele verursacht, zu zeigen als Ziel gesetzt. Riassunto. L’artcolo è sorto durante la nostra permanenza a Bodø in Norvegia per il progetto Socrates Comenius 2.1 come reazione ad un eccessivo uso delle calcolatrici nelle lezioni di matematica. L’esperimento è stato realizzato in due classi norvegesi ed una slovacca. Il nostro obiettivo è stato quello di mostrare che un eccessivo uso delle calcolatrici può causare la riduzione dell’abilità di valutazione dei risultati di semplici compiti matematici, rispettivamente un esatto calcolo dei compiti matematici.
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This article was partly supported by project Socrates Comenius 2.1 n. 106663-CP-1-2002-1-ITCOMENIUS-C21 and by European Social Funds JPD 3 BA-2005/1-063 and SOPLZ-2005/1- 225
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Abstrakt. Článok vznikol počas pobytu v nórskom Bodø počas projektu Socrates Comenius 2.1 ako reakcia na prílišné využívanie kalkulačky na hodinách matematiky. Experiment prebehol v dvoch nórskych triedach a jednej slovenskej. Našim cieľom bolo pokúsiť sa ukázať, že používanie kalkulačiek môže zapríčiniť zníženie schopnosti odhadu výsledkov jednoduchých matematických úloh, resp. presného výpočtu matematických úloh. Key words: experimental analysis, estimation of results, simply mathematical tasks, using of calculators on mathematical lessons
1 INTRODUCTION Arose during our stay in Norwegian Bodø during The Socrates Comenius 2.1 Project we participated on lessons of mathematic in secondary school. We remark excess using calculator in educational process by students even on first lessons. We were surprised because of calculating 2*2 on calculator. This experience gives us an idea acknowledges their abilities made simple calculation operation without using calculators. With simple calculation we mean tasks, which need for solution only basic mathematical abilities (basic mathematical abilities = addition, subtraction, multiplication, division integers and decimal). We take a specimen of some students on the first level in secondary school in Norway, and after our return home we made research on Slovak school for comparing. We divide our research into two parts: to find out their abilities to make estimation in multiplication, addition and dividing (max. numbers with three digit) to find out their abilities to make correct calculation with using only basic mathematical abilities For a consideration of write problems we make this goal: C: To detect if there exist some differences between the abilities of the students in the same level of psychical evolution to make exact calculation, eventually to estimate result of basic mathematical task and if these differences can be related with the using of calculators at Math lessons.
2 PREPARATION OF THE PROJECT We were preparing this project by preparation lessons in Bodø in Norway, where we stayed on stay. We prepared two sets of tasks – first one was about estimation (U1), second one was oriented to exact calculation (U2). Both sets could be solved using only basic mathematical operation. There are tasks, which were supposed to solve:
PROBLEMS WITH USING CALCULATORS ON MATHEMATICAL LESSONS
U1: Try to estimate as well as possible: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
21 ⋅ 33 42 ⋅ 89 0, 7 ⋅ 0, 21 13 + 79 22,9 + 56 + 93,8 3 ⋅ 12 + 2 ⋅ 34 2 ⋅ 18,1 + 3 ⋅ 98,9 0, 29 ⋅ 0,3 717 : 7 12 ⋅ 1, 20
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U2: Calculate: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
−70 + 60 −12 + 35 −32 − 49 12 − ( +5 ) − ( −6 ) −6 + ( −5 ) + ( −9 ) −6 − ( −10 ) + ( −2 ) −2,5 − ( + 3,1) + 4 −9 − ( −7 ) + ( −3) − ( −2 ) + 8 −2 ⋅ 7 ⋅ 5 ⋅ ( −3) 24 : 4 + 6 ⋅ ( −2 ) + 8 ( 5 − 13) : 2 ⋅ ( −5) ⋅ ( 3 − 5)
We didn’t have enough time to realize this experiment in more than one classes, but the first part we did in two Norwegian classes. 2.1 ANALYSIS A PRIORI OF ESTIMATION We assumed that the students would be solving tasks using adequate rounding. They could make these mistakes: U1-1: 21 ⋅ 33 • Incorrect place value of result of multiplication, for example 60, 6000 U1-2: 42 ⋅ 89 • Incorrect place value of result of multiplication • Incorrect rounding of number 89 to number 80. Estimation with this rounding will not by very good U1-3: 0,7 ⋅ 0, 21 • Error in place for decimal point • Incorrect rounding, for example 1·0,21 U1-4: 13 + 79 • Numerical mistake U1-5: 22,9 + 56 + 93,8 • Numerical mistake U1-6: 3 ⋅ 12 + 2 ⋅ 34 • Numerical mistake by multiplication • Numerical mistake by addition • Incorrect rounding, for example number 34 to number 40 U1-7: 2 ⋅ 18,1 + 3 ⋅ 98,9 • Numerical mistake by multiplication • Numerical mistake by addition
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• Incorrect rounding U1-8: 0, 29 ⋅ 0, 3 • Error in place for decimal point, for example 0,9 • Incorrect rounding U1-9: 717 : 7 • Incorrect place value of result of division, for example 10 • Numerical error U1-10: 12 ⋅ 1, 20 • Incorrect place value of result, for example 1,44 or 144 • Numerical error by multiplication • Incorrect rounding, for example 12·1 2.2 ANALYSIS A PRIORI OF EXACT CALCULATION U2-1: – 70 + 16 • Numerical error • Error in sign of result U2-2: –12 + 35 • Numerical error • Error in sign of result U2-3: –32 – 49 • Numerical error • Error in sign of result U2-4: 12 – (+ 5) – (– 6) • Numerical error • Error in sign of result • Error in eliminating of bars U2-5: – 6 + (– 5) + (– 9) • Numerical error • Error in sign of result • Error in eliminating of bars U2-6: – 6 – (– 10) + (– 2) • Numerical error • Error in sign of result • Error in eliminating of bars U2-7: – 2,5 – (+ 3,1) + 4 • Numerical error • Error in sign of result • Error in eliminating of bars • Error in addition of decimal number
PROBLEMS WITH USING CALCULATORS ON MATHEMATICAL LESSONS
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U2-8: – 9 – (– 7) + (– 3) – (– 2) + 8 • Numerical error • Error in sign of result • Error in eliminating of bars • Vacation because of long notation U2-9: –2 ⋅ 7 ⋅ 5 ⋅ (– 3) • Numerical error • Error in sign of result U2-10: 24 : 4 + 6 ⋅ ( – 2) + 8 • Numerical error • Error in sign of result • Error in priority of operation • Error in eliminating of bars U2-11: (5 – 13) : 2 ⋅ (– 5) ⋅ (3 – 5) • Numerical error • Error in sign of result • Error in priority of operation • Error in eliminating of bars • Error in strategy of solving
3 THE REALIZATION OF EXPERIMENT – CLASS 1MX (NORWAY) We were realized our experiment at the beginning of 3rd lesson. The students knew about little test and they could decide if they’d like to participate on it. These of them, which didn’t have interest of participating could stayed in class and did something else in that way not to disturb their classmates.
First part: We gave away the students papers, where they had to write the numbers of calculations from 1 to 10 one under another. As follows we explained them what we expected from them (in English, but the teacher repeat it in Norwegian language to make sure that they understood). Students had 15 seconds to solve each tasks. Zuzana was “a Second”; Majka was “a Recorder”. Students had to write only results. At the beginning two tasks were written onto the blackboard. First was shown the students to estimate, second was hidden. After 15 seconds the first one was given out and the second one was released. The third one was written onto the black board and hidden. I ran this way till the end of the test.
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Second part: After compiling papers with results we gave away the students the other set of tasks. They had 20 minutes to solve. Some students delivered us their results after 7 minutes the rest of them used all time. After finishing the second part we thanked students for their readiness to be participated at this experiment.
3.1 THE EVALUATION OF THE EXPERIMENT The results of the first part (Quantitative analysis a posteriori): Table 1 O1 s1 0 s2 1 s3 0 s4 1 s5 1 s6 1 s7 1 s8 1 s9 1 s10 1 s11 1 s12 1 s13 1 s14 1 s15 0 s16 1 s17 0 s18 1 Average: 4,88
O2 0 0 1 0 1 n 1 0 0 0 1 0 1 n 0 0 0 1
O3 0 0 1 0 1 0 0 n 1 1 n 0 0 n 1 1 0 0
O4 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0
O5 1 1 0 1 1 n 0 1 1 1 n 1 1 1 0 0 0 1
O6 0 0 1 0 0 0 0 0 0 n 0 0 0 0 0 0 0 0
O7 0 1 1 0 1 n 0 n 0 0 0 1 1 0 n 1 0 0
O8 0 0 0 0 0 0 n n 0 0 n 1 0 n 0 0 0 0
O9 O10 Together 0 1 3 0 1 5 1 1 7 1 1 5 0 1 7 1 1 4 1 1 5 0 n 3 1 1 6 1 1 6 n 1 3 1 1 7 0 1 6 1 1 5 1 1 4 0 1 5 1 1 2 1 1 5
Additional text: valid for the charts 1−5 Si − … marking the students Oi – … marking the estimations Ui − … marking the tasks
0 – false estimation 1 – right estimation n – without answer
PROBLEMS WITH USING CALCULATORS ON MATHEMATICAL LESSONS
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Qualitative analysis a posteriori: As it is seemed from the chart, some students had more problems with estimation. The most problematic estimation was found to be O6, where was only one right answer and O8, where was also only one right answer and 4 students didn’t even solve it. At O6 we hadn’t expected so big problem in solution. Task O8 we had considered for one of the most difficult one, what also confirmed us. The estimation O10 was solved the best, as only one student didn’t answer and the rest of them answered right. „The test“ from estimation was written on the base of spontaneity. Some of the students wanted to know the results of their aiming. Those, who didn’t care of prestige, didn’t try to solve some tasks or they didn’t catch it to the time limit. The teacher and also the students liked this form of checking and it probably will be included as the diversity of the lesson.
The results of the second part (Quantitative analysis a posteriori): Table 2 U1 s1 1 s2 1 s3 1 s4 1 s5 1 s6 1 s7 0 s8 1 s9 1 s10 1 s11 1 s12 1 s13 1 s14 1 s15 1 s16 1 s17 1 s18 0 s19 1 Average: 8,84
U2 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1
U3 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1
U4 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1
U5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1
U6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
U7 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 1
U8 0 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1
U9 U10 U11 Together 0 0 0 6 1 1 0 10 1 0 1 10 0 1 1 10 1 1 1 11 1 1 1 11 1 0 1 9 1 1 1 11 1 1 n 9 1 0 0 7 1 1 0 10 1 0 1 10 1 1 1 6 1 1 0 9 1 0 n 8 1 1 0 10 1 0 0 8 0 0 0 4 1 1 0 9
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Qualitative analysis a posteriori: Except the student s18 all of them had more than half of tasks solved in correct way. We think that they made rather basic mistakes in their role as the first class students. The reason could be extreme ambition to write the „test“ as the best, or on the other hand – as it won’t be the part of evaluation, they could write there what they wanted. Some of students had problem to count without calculator, what could be seen on their faces.
4 THE REALIZATION OF THE EXPERIMENT – CLASS 1HS (NORWAY) We implemented this experiment into our main project. Because of absence of the time we could only implement the first part. This part swept at the second lesson. We went the same way as in the before mentioned class. The differences between the classes were in the interest of the students in mathematics and in the amount of math lessons per week.
4.1 THE EVALUATION OF THE EXPERIMENT Quantitative analysis a posteriori: Table 3 O1 s1 1 s2 1 s3 1 s4 0 s5 0 s6 n s7 n s8 n Average: 2,75
O2 0 0 0 0 0 n n n
O3 0 0 0 0 0 n n n
O4 1 0 1 1 0 1 1 1
O5 0 1 1 1 0 n n 0
O6 n 1 1 1 0 n n n
O7 1 0 1 0 0 n n n
O8 0 0 0 0 0 n n n
O9 O10 Together 1 1 5 1 1 5 0 1 6 0 0 3 n n 0 n n 1 n n 1 n n 1
Qualitative analysis a posteriori: The students were taken by surprise with the test. At the lesson they were on „the area of plane shapes“ and we wrote the test at the end of the lesson. Each student sat alone at the desk. The students were confused first by the time
PROBLEMS WITH USING CALCULATORS ON MATHEMATICAL LESSONS
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limit they had to make estimation, what manifested also on results (first three bars are – accepting simplicity of examples – taken by 1 only small). The estimation of O8 was problem also in this group – nobody solved it right. Together with O2 and O3 they are the most problematic tasks, either the result of these two ones are influenced by the time limit, which students didn’t like.
5 THE REALIZATION OF THE EXPERIMENT − CLASS SEXTA A (SLOVAKIA) To make comparison we carried out the experiment at the Slovak school with the students of the same age. The experiment was carried out at the Secondary school of A. Bernolak in Námestovo. The students of chosen class aren‘t specialist in mathematics and any other subject. They have general direction. The experiment went on similar as in the class in Norway. A teacher provided enough time, so both parts of experiment could be overrun. All students took part on this experiment, although it was voluntary.
5.1 THE EVALUATION OF THE EXPERIMENT The results of the first part (Quantitative analysis a posteriori): Table 4 s1 s2 s3 s4 s5 s6 s7 s8 s9 s 10 s 11 s 12 s 13 s 14 s 15 s 16
O1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1
O2 1 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0
O3 0 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0
O4 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1
O5 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1
O6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
O7 1 1 0 0 1 1 0 0 0 0 1 1 0 0 0 n
O8 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 n
O9 O10 Together 1 1 8 1 1 7 0 1 5 0 1 4 1 1 7 1 1 9 1 0 5 1 1 7 1 1 6 1 1 6 1 1 8 1 1 9 1 1 7 1 1 8 0 0 4 0 1 5
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s 17 1 s 18 1 s 19 1 s 20 1 s 21 1 s 22 1 s 23 1 s 24 0 s 25 1 s 26 1 s 27 1 Average: 6,26
0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1
1 1 1 1 1 1 1 0 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 0 0 1 1 0
0 1 1 0 0 1 0 0 1 1 1
1 0 0 0 0 0 0 0 1 0 0
1 0 0 0 1 1 0 0 1 1 1
1 1 0 1 1 1 0 1 1 1 1
7 6 5 5 7 7 3 2 8 7 7
Qualitative analysis a posteriori: We can see that the most problematic task was O8 again, but students made a lot of mistakes also in O2 and O3. The reason could be initial stress (similar as in 1HS), insufficient concentration at the beginning, or one of the mistakes described in analysis a priori. The students reached very nice average, but those one who had less than half of estimation correct (for example s23, s24) should train more in estimation.
The results of the second part (Quantitative analysis a posteriori): Table 5 U1 s1 1 s2 1 s3 1 s4 1 s5 0 s6 1 s7 1 s8 0 s9 1 s 10 1 s 11 1 s 12 1 s 13 1 s 14 1
U2 1 1 1 1 1 1 1 1 1 1 1 1 1 1
U3 0 1 1 1 1 1 1 1 1 0 1 1 1 1
U4 1 1 1 1 1 1 1 1 1 1 1 0 1 1
U5 0 0 1 1 0 1 1 1 1 1 1 1 1 1
U6 1 1 1 1 0 1 1 1 1 1 1 1 1 1
U7 1 1 1 1 1 1 1 1 1 1 1 0 1 0
U8 1 1 1 1 0 1 1 1 1 1 1 1 1 1
U9 U10 U11 Together 0 1 0 7 1 1 1 10 1 1 1 11 1 1 1 11 1 1 1 7 1 1 1 11 1 1 1 11 1 1 1 10 1 1 1 11 1 1 1 10 1 1 1 11 1 1 1 9 1 0 0 9 1 1 1 10
PROBLEMS WITH USING CALCULATORS ON MATHEMATICAL LESSONS
s 15 1 s 16 1 s 17 1 s 18 1 s 19 1 s 20 1 s 21 1 s 22 1 s 23 1 s 24 1 s 25 1 s 26 1 s 27 1 Average: 9,96
1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 0 1 1 1 1
0 1 1 1 1 0 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 0 1 1
0 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 0 0 1 1 1 1
1 1 1 1 1 0 1 0 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 0 0 1 1 1
11
8 11 11 11 11 9 11 9 8 10 10 11 11
Qualitative analysis a posteriori: The students have reached beautiful result in exact calculations what shows the necessity of exact results at the math lessons. The most problematic tasks were U5, U7 and U11, where decimal numbers and combination of different arithmetical operations could also influence the difficulties.
How did we evaluate: Part 1: We define an acceptable interval for each example, which we accepted results from. Part 2: We accepted only right answers.
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6 CONCLUSION The above brought charts show us, that the students of Slovak school, who use calculators less often, were more able in mathematics – calculations than the students from Norwegian school, who use calculators very often even from lower class of primary school. There wasn’t our effort in this article to bring a proof that using of calculators at the math lessons attenuates the ability of the students to reach the correct results without them (for the first – the pattern wasn’t big enough, for the second – not only the using of calculators influents the ability of the math skills). Our goal was to show that there is a problem like this and it’s value to explore it and to alert the teacher who maintain the trends of the teaching methods of the west of Europe, or USA, that these methods needn’t bring improvement, but can also bring some negative effects. If each student has a calculator, the necessity of practice of some manual calculation will abandon. On the other hand there could be seen some facts we can’t see now. For example it can be shown that the practice of the algorithm of adding is good propedeutics to practice and understanding of the algorithm as themselves. The possible eliminating of this practice should be by passed with another, for example with the practicing of estimation.
REFERENCES Bereková H., Főldesiová L., Regecová M., Kremžárová L., Slavíčková M., Trenčanský I., Vankúš P., Zámožiková Z.: Slovník teórie didaktických situácií, časť 2. In: Zborník 5 Bratislavského seminára z teórie vyučovania matematiky, Bratislava, Univerzita Komenského 2003, str. 113–122 Feketeová T., Holá K., Vankúš P., Vinceková E.: Nórsky školský systém. In: Zborník 6 Bratislavského seminára z teórie vyučovania matematiky, Bratislava, Univerzita Komenského 2004, str. 39–46 Hejný M.: Teória vyučovania matematiky II, SPN Bratislava 1990, str. 68 Slavíčková M.: Didaktické problémy pri základných aritmetických operáciach. Rigorózna práca, október 2004, str. 20 Spagnolo, F.: La recherche en didactique des mathématiques: un paradigme de référence. Zborník príspevkov na seminári z teórie vyučovania matematiky, Bratislava, 1999
ZUZANA BRISUDOVÁ, Department of Algebra, Geometry and Didactics of Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia E-mail:
[email protected] MÁRIA SLAVÍČKOVÁ, Department of Algebra, Geometry and Didactics of Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia E-mail:
[email protected] 
