59中英文雙語數(shù)學(xué)教育專業(yè)外文文獻翻譯成品中學(xué)數(shù)學(xué)教學(xué)中的向量計算以及立體幾何

1、<p>　　外文標(biāo)題：VECTOR CALCULUS AND SOLID GEOMETRYAT TEACHING OF MATHEMATICS AT SECONDARY SCHOOL</p><p>　　外文作者：LUCIA RUMANOVÁ </p><p>　　文獻出處: Acta Didactica Universitatis Comen

2、ianae Mathematics,Issue6,2006 </p><p>　　英文2389單詞， 12963字符，中文3702漢字。</p><p>　　此文檔是外文翻譯成品，無需調(diào)整復(fù)雜的格式哦！下載之后直接可用，方便快捷！只需二十多元。</p><p>　　VECTOR CALCULUS AND SOLID GEOMETRYAT T

3、EACHING OF MATHEMATICS AT SECONDARY SCHOOL</p><p>　　LUCIA RUMANOVÁ</p><p><b>　　Abstract</b></p><p>　　Paper is presentation analysis of problem in the theory of dida

4、ctic situation and the evaluation of experiment with using statistical program CHIC. Problem analysis is task of stereometry for students from secondary schools. Aim of experiment was mention on problems with application

5、 of knowledge of different part of mathematics at solution of stereometry problem.</p><p>　　Key words: theory of didactic situations, analysis a priori, analysis a posteriori, statistical program C.H.I.C., s

6、olid geometry, synthetic geometry, vector calculus, analytic geometry</p><p>　　INTRODUCTION</p><p>　　One of the long – lasting problems in teaching mathematics at secondary school is the problem

7、 of relations among subjects, respectively the continuity of mathematics content and the content of other subjects. However, a specific prob- lem is the continuity or the co-ordination of individual parts of mathematics

8、in the process of education. In this work I want to pay attention to this part of teach- ing mathematics, specifically geometry ? focus on the relation between solid geometry from the vec</p><p>　　We know th

9、at the pupils are taught the basics of vector algebra and analytic geometry already in the 8th or 9th grade at the primary school. Later on they get other information with the notion of vector and its operations (sum, od

10、ds...) in the first grade at secondary school, and this part is mostly used as a tool for knowledge analytic geometry, for example equations of the straight lines, planes, ... and later, for the analytic geometry of coni

11、cs, balls, and so on.</p><p>　　Another problem is little attention paid to application of the vector calculus to solve the tasks of solid geometry. These tasks are taught in other parts only by the synthetic

12、 geometry, as well as little time spent for application tasks in other subjects (physics, geology, geography, etc.). Students usually don’t find relative with other subjects at solution of solid problems whereupon their

13、know- ledge they don’t know or don’t tried to apply.</p><p>　　AIMS OF RESEARCH</p><p>　　The goal of research is therefore emphasize problems of students at secondary school with solving solid pr

14、oblems, with application their knowledge (from differ- ent part of mathematic) in solid geometry, as well as mention problems of students to find connection between parts of mathematics and other learning subjects.</p

15、><p>　　Then the aim of research is also experimental verify validity of the hypothe-</p><p>　　ses H1–H4 and eventually suggest solutions of effectiveness education of analytical (vector) geometry w

16、ith relating parts of mathematic and thereafter enhancement education between learning subjects at secondary school.</p><p>　　Abovementioned problems we summarized to following hypotheses:</p><p&g

17、t;　　H1 The solid geometry is taught at secondary school separately, i. e. students are separately taught axiomatic, separately deal with synthetic geometry, separately analytic geometry, and so the sequence of various ap

18、proaches is minimal or any.</p><p>　　H2 In mathematics textbook there aren’t examples with unification character, which would promote elimination limitations (to fault) listed in H1.</p><p>　　H3

19、 Students of secondary school with the established educational schedule ha- ven’t enough ability to apply their knowledge of the vector calculus in other areas of mathematics, apart from analytic geometry, and that in th

20、is case only formally.</p><p>　　H4 Students of secondary school are not able in ample measure to be aware of the continuity of synthetic and analytic geometry (vector calculus) in solution of particular prob

21、lem situations. Similar situation applies to University students who will be teachers of mathematics.</p><p>　　PREPARATION OF THE EXPERIMENT</p><p>　　In accordance with the tenets theory of dida

22、ctic situations frame: within the frame of the didactic situation S3 (noosferic didactic situation) we made an analy- sis of math textbooks for secondary schools, an analysis of various mathematical materials, where the

23、goal was to choose a useful problem for students and which would help us to find out reply to already formulated hypothesis H1, H2, H3 and H4 in the introduction. Our goal was to seek such a problem, which the students w

24、ere not able to </p><p>　　The task for the students in this experiment was to try giving an example from the solid geometry with exploitation knowledge out of analytic geometry and vector calculus. (What we

25、awaited results). Results of experiment have mention problems with solving solid problems, with application their knowledge (from different part of mathematic) in solid geometry, as well as mention problems of students t

26、o fin connection between parts of mathematics and other learning subjects. Results of this experime</p><p>　　The task was:</p><p>　　Given is a cube ABCDEFGH and K-point, L-point, M-point, N-poin

27、t, so that K-point is centre of upper surface EFGH, L-point is centre of the AB, M- point belong to AE, where,and N-point belong to BG.Are the points K, L, M, and N coplanary?</p><p>　　Based on the above-me

28、ntioned criteria the final sentence of the given task can be interpreted in several different ways, their experimental attesting will be the part of our following research of this field.</p><p>　　Q1: Vector

29、calculus ? exploitation collinearity of vectors or coplanarity</p><p>　　of vectors</p><p>　　Q2: Analytic solution ? writing general equation of plane assigned three</p><p>　　from fo

30、ur given points and then we prove incidence of fourth point into plane</p><p>　　Q2´: Analytic solution ? writing parametric equation of plane assigned</p><p>　　three from four given points

31、and then we prove incidence of fourth point into plane</p><p>　　Q2´´: Analytic solution ? writing parametric equations of two lines from</p><p>　　given fourth points and then we fount

32、out their relative position (if they construct of plane)</p><p>　　Q3: Synthetic approach ? construction section of plane of cube and then we</p><p>　　prove incidence fourth point into plane</

33、p><p>　　Q4: Vector calculus ? exploitation attributes of “barycentre</p><p>　　POSSIBLE STRATEGIES OF STUDENTS SOLUTION – ANALYSE A PRIORI OF PROBLEM DESIGNATION TO EXPERIMENT</p><p>　　

34、we construct the section plane LMN of cube ABCDEFGH and than</p><p>　　we attest: ? K ∈LMN ? (That they are single calculations – for examples: solution with exploitation follows Figure 2 or exploitation rese

35、mblance be- tween triangle)</p><p>　　And then the question is:</p><p>　　We solve the system of equations:</p><p>　　For parameters t ?????and s ???2 , the equations have a solution,

36、there out resulting</p><p>　　The problem is: Are the vectors LS and LK collinear? (point S is a centre</p><p>　　of line segment MN ). If the vectors are collinear, so the K-point, L-point, M-<

37、;/p><p>　　point, N-point are coplanar.</p><p>　　For the vector LS resulting these terms:</p><p>　　By means of substitution relations and simple reforms resulting term:</p><p

38、>　　K-point is a barycentre of E(1), G(1), L-point is a barycentre of A(2), B(2), M-point is a barycentre of A(2), E(1) and N-point is a barycentre of B(2), G(1).</p><p>　　And then, the question is: What i

39、s barycentre G of this points set {A(2),</p><p>　　B(2), G(1), E(1)} ?</p><p>　　From the facilities of barycentre ? G is barycentre of {M(3), N(3)} and{K(2), L(4)}</p><p>　　Analysis

40、a-priori problem had been formulated and teacher’s activity in a-didactic situation</p><p>　　The base of didactic research is the test students´ reactions in the given di- dactic situation. So as we cou

41、ld make the test to realize at the first we had to do the full analysis of didactic situation itself.</p><p>　　Part of the experiment is the integration individual phases of the problem</p><p>　

42、　solution to system of levels in the analyse didactic situation.</p><p>　　This is the tablet composition of milieu and with it associated types of didac- tic situations (Margolinas 1994).</p><p>

43、;　　ANALYSE OF TEACHER’S WORK (DESCENDING ANALYSIS)</p><p>　　S3 – noosferic situation – on this stage we analysis the math textbook for sec- ondary school (2nd or 3rd class), analysis various mathematical mat

44、erials (educa- tional schedule…), specifically study of solid geometry, vector algebra and ana-</p><p>　　lytic geometry. We want to choose problem, which the students were not able to</p><p>　　s

45、olve with the learnt simple algorithms and students can use knowledge from different parts of mathematics and other learning subjects. Finish of noosferic situation will be the milieu for the next situation.</p>&

46、lt;p>　　S2 – constructional situation – teacher will try to find examples, that were defined in noosferic situation S3 and on the other side in situation S1, in which they will be able to realize. Goal was to choose a

47、useful problem for students which would help him to find reply to already formulate hypothesis H1, H2, H3 and H4. They are examples which students can abet in examples solution Q1, Q2, Q2´, Q2´´, Q3 a Q4.&

48、lt;/p><p>　　S1 – project situation – in situation S1, teacher writes a text of the example</p><p>　　and he “projects” his solution. Student is one on teacher’s consciousness. This is a situation, w

49、hich involves student’s activity, too. The student can solve prob- lem in a way that he constructs the section plane of cube, and then he finds out if other point is point of plane; or the student solves a problem that h

50、e writes parametric equation of plain and he finds out if the fourth point is the point of the plane; or he applies exploitation collinearity of vectors or coplanarity or exploita- ti</p><p>　　S0 – didactic

51、situation – in this situation we analysis and do institutionalisa- tion of the new knowledge and we formulate the problem. Teacher takes care of</p><p>　　designed goals and he follows student’s solution, too

52、. It is a situation where the</p><p>　　analysis of teacher’s work and analysis of student’s work meet, and the didactic situation will be the result of the teaching process</p><p>　　CONCLUSION&l

53、t;/p><p>　　Problem at teaching solid geometry is sequence single parts of mathematics within solid geometry. With this problem relate too solutions of solid geometry problems. Concrete, students have problem so

54、lve problem, students could solve problem in different theoretical frame with reference to their level of knowledge and sciential level at secondary school, but which the students were not able to solve problem only with

55、 the learnt simple algorithms.</p><p>　　The goal of research is mention problems with application students´ knowledge at solving problem from different part of mathematic in solid geometry and also expe

56、rimental verify validity or invalidity of the hypotheses with using peda- gogical experiment.</p><p>　　Experiment was realized also in university (future teachers of mathematics) and confirmed us that there

57、is similar problem with application students´ knowl- edge from different parts of mathematics in solid geometry and also from other subjects. Whereupon, teacher prefer concrete parts in teaching (in lessons of mathe

58、matics) and so teacher often disuse examples with unification character.</p><p>　　Conclusion of the work includes collections of the tasks and we want (of course) expand any interesting application problems

59、(tasks) from solid geome- try, which we can improve on lessons of mathematics. Consequently, correction of collections of these tasks for practical use as part of texts in books of mathe- matics for secondary schools. In

60、 continues of experiment we want to try making educational texts some parts of solid geometry. Educational texts will be especially orientated to coordinat</p><p>　　REFERENCES</p><p>　　Balacheff

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76、 nella scuola secondaria, Palermo, 1998</p><p>　　Spagnolo F.: La recherche en didactique des mathématiques: un paradigme de référence, Zborník</p><p>　　bratislavského se

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79、órie didaktických situácií na zefektívnenie u?enia sa, Zborník bratislavského</p><p>　　seminára z teórie vyu?ovania matematiky, 2001.</p><p>　　中學(xué)數(shù)學(xué)教學(xué)

80、中的向量計算以及立體幾何</p><p>　　LUCIA RUMANOVÁ</p><p><b>　　摘要</b></p><p>　　在本文中，呈現(xiàn)了對教學(xué)情境理論中的問題和對CHIC使用統(tǒng)計程序的實驗評估的分析。 分析的問題是有關(guān)于中學(xué)立體幾何。 實驗的目的是在數(shù)學(xué)不同部分知識中去解決立體幾何的應(yīng)用問題。</p>

81、<p>　　關(guān)鍵詞：理論情境、實驗前分析、試驗后后分析、統(tǒng)計程序C.H.I.C、立體幾何、綜合幾何、向量計算、解析幾何</p><p><b>　　引言</b></p><p>　　在中學(xué)數(shù)學(xué)教學(xué)中，長期存在的問題之一是學(xué)科間的關(guān)系問題，分別是數(shù)學(xué)內(nèi)容的連續(xù)性和其他學(xué)科內(nèi)容的問題。 然而，一個具體的問題是數(shù)學(xué)的各個部分在教學(xué)過程中的連續(xù)性或協(xié)調(diào)性。 在這項工

82、作中，我想關(guān)注數(shù)學(xué)教學(xué)的這一部分，特別是幾何學(xué)，重點關(guān)注向量計算、立體幾何與立體幾何教學(xué)方法之間的關(guān)系。</p><p>　　眾所周知，小學(xué)生在八年級或九年級已經(jīng)被教授過向量代數(shù)和解析幾何的基礎(chǔ)知識。 后來，他們在中學(xué)一年級學(xué)習(xí)向量及其運算的概念，并且這部分主要用作解析幾何知識學(xué)習(xí)的工具，例如直線方程線條，平面計算......以及后來的錐形、球形等解析幾何。</p><p>　　另一個很少

83、關(guān)注的問題是向量計算的應(yīng)用以及在立體幾何問題解決的應(yīng)用。 這些任務(wù)只在其他部分以綜合幾何學(xué)的方式教授，而其他科目（物理學(xué)，地質(zhì)學(xué)，地理學(xué)等）對該方面的應(yīng)用很少提及。 學(xué)生通常沒有找到該學(xué)科與其他學(xué)科的關(guān)系，發(fā)現(xiàn)這些關(guān)系有利于解決他們不知道或不會的立體幾何問題。</p><p><b>　　研究目標(biāo)</b></p><p>　　因此，此次研究的目標(biāo)是強調(diào)中學(xué)生解決立體幾

84、何的問題，并將他們的知識（來自數(shù)學(xué)的不同部分）應(yīng)用于立體幾何中，本次研究還會提及學(xué)生發(fā)現(xiàn)數(shù)學(xué)部分與其他學(xué)習(xí)科目之間的聯(lián)系。 之后的研究先假設(shè)H1-H4的有效性并進行實驗驗證假設(shè)，最終提出向量幾何以及與數(shù)學(xué)相關(guān)部分在教育解決方案上的有效性，以提升中學(xué)的學(xué)習(xí)科目之間的聯(lián)系。</p><p>　　根據(jù)上述問題我們總結(jié)為以下假設(shè)：</p><p>　　H1 立體幾何學(xué)是分別都是在中學(xué)教授的，即 學(xué)

85、生被分開去教授公理化，分別去處理綜合幾何、單獨分析幾何，所以各種方法的序列最小或者以任何其他方式排列。</p><p>　　H2在數(shù)學(xué)教科書中沒有具有統(tǒng)一性的例子，這將有利于H1中列出的問題的解決。</p><p>　　H3在學(xué)校按照作息時間上學(xué)的中學(xué)學(xué)生除了解析幾何外，還沒有足夠的能力將其在數(shù)學(xué)的其他領(lǐng)域的應(yīng)用知識應(yīng)用于向量計算，這種情況只會發(fā)生在學(xué)校。</p><p

86、>　　H4中學(xué)的學(xué)生無法充分地了解綜合和分析幾何（向量運算）的連續(xù)性</p><p>　　特定問題的情況。 類似的情況也適用于將來會成為數(shù)學(xué)老師的大學(xué)學(xué)生。</p><p><b>　　實驗的準(zhǔn)備工作</b></p><p>　　根據(jù)教學(xué)理論的教學(xué)情境框架：在教學(xué)情境S3（新教學(xué)情境）的框架內(nèi)，我們對中學(xué)的數(shù)學(xué)教科書進行了分析，分析了各種

87、數(shù)學(xué)材料，其目標(biāo)是對學(xué)生來說選擇一個有用的問題，并且可以幫助我們找到對引言中已經(jīng)提出的假設(shè)H1，H2，H3和H4的回應(yīng)。 我們的目標(biāo)是尋找這樣一個問題，學(xué)生們無法用簡單的算法解決問題，在數(shù)學(xué)教科書中通常不存在這些類型的問題。</p><p>　　在這個實驗中，學(xué)生的任務(wù)是嘗試從已經(jīng)學(xué)習(xí)的立體幾何知識學(xué)中解析出幾何和向量演算。 （這正是我們期待的結(jié)果）。 實驗結(jié)果提出了解決立體幾何問題的方法，并將他們的知識（來自數(shù)

88、學(xué)的不同部分）應(yīng)用于立體幾何中，并提出學(xué)生將數(shù)學(xué)部分與其他學(xué)習(xí)科目聯(lián)系起來的問題。 這個實驗的結(jié)果必須說明學(xué)生如何使用這些計算單位中獲得滿足感。</p><p><b>　　任務(wù)是:</b></p><p>　　已知立方體ABCDEFGH 和點K, 點L,點 M,點 N, 點K在立體圖形上面EFGH的中心位置, L點上線段AB的中點, 點M AE上, ,點 N 在線段

89、BG上，.那點 K, L, M, 和點 N 共面嗎?</p><p>　　根據(jù)上述標(biāo)準(zhǔn)，已知任務(wù)的最后一個要解答的問題可以用幾種不同的方式來解釋，他們的實驗證明將成為我們對這一領(lǐng)域后續(xù)研究的一部分。</p><p>　　問題1：向量計算--向量的共線性或向量的共面性知識的提煉</p><p>　　問題2：解析解 - 寫出從四個已知點分配三個平面的一般平面方程，然后證

90、明第四個點進入平面的入射角。</p><p>　?。?）問題2：解析解 - 寫出從四個已知點分配三個平面的平面參數(shù)方程，然后證明第四個點進入平面的入射角。</p><p>　　（2）問題2：解析解- 從已知的第四點寫出兩線段的參數(shù)方程，然后我們找出它們的相對位置（如果它們是平面的構(gòu)造）</p><p>　　問題3：綜合方法 - 立方體平面截面，然后證明第四點的入射角

91、。</p><p>　　問題4：向量計算---“重心屬性”的提煉</p><p>　　4學(xué)生解決問題可能出現(xiàn)的策略 - 實驗中分析指定問題的先決個前提</p><p>　　我們構(gòu)造ABCDEFGH立方體的剖面LMN</p><p>　　b）我們證明：？K∈LMN？ （它們是單一計算 - 例如：利用圖2的解決方案或三角形之間的相似性）。<

92、/p><p><b>　　問題是：</b></p><p><b>　　列出方程式就是:</b></p><p>　　參數(shù)中 t=3/2 ?， s ?=??2 , 方程有解，結(jié)果是</p><p>　　問題是：向量LS和LK是否共線？ （點S是線段MN的中心）。 如果向量是共線的，那么K點，L點，M點，

93、N點是共面的。</p><p>　　向量LS會帶來下面的關(guān)系：</p><p>　　通過替代關(guān)系和簡單的替換得到下面的關(guān)系式：</p><p>　　K點是E（1），G（1）的重心，L點是A（2），B（2）的重心，M點是A（2）的重心，E（1） N點是B（2），G（1）的重心。</p><p>　　然后，問題是：這個點集的重心是什么？是{A（2

94、），B（2），G（1），E（1）}？</p><p>　　重心的前提條件?G是{M（3），N（3）}和{K（2），L（4）}的重心</p><p>　　實驗之前的問題已經(jīng)得到表述以及教師在教學(xué)情境中的教學(xué)活動</p><p>　　教學(xué)研究的基礎(chǔ)是測試學(xué)生在已知情景中的反應(yīng)。 因此，我們可以在第一時間進行測試，我們必須對教學(xué)情況本身進行全面分析。</p>

95、<p>　　部分實驗是在分析教學(xué)情境中，將問題解決方案的各個階段整合到各層次系統(tǒng)中去。</p><p>　　這是平面構(gòu)成和與其相關(guān)的情況說明類型（Margolinas 1994）。</p><p>　　教師工作分析（減少分析的部分）</p><p>　　S3 - 創(chuàng)新情境- 在這個階段我們分析數(shù)學(xué)教科書（二年級或三年級），分析各種數(shù)學(xué)材料（教育計劃...

96、），具體研究立體幾何、向量代數(shù)和解析幾何。我們想要選擇問題，學(xué)生不能用學(xué)習(xí)過的簡單算法解決問題，學(xué)生可以使用數(shù)學(xué)和其他學(xué)習(xí)科目的不同部分的知識。創(chuàng)新情境的完成將為下一情境做好鋪墊。</p><p>　　S2 - 構(gòu)造情境 - 教師將嘗試尋找例子，這些例子在S3的創(chuàng)新情境中有定義，另一方面也在情境S1中去定義，他們將能夠認識到這些例子。目標(biāo)是為學(xué)生選擇一個有用的問題，幫助他找到已經(jīng)提出假設(shè)的H1，H2，H3和H4四

97、個問題的答案。他們是學(xué)生們可以利用的例子，即Q1，Q2，Q2'，Q2“，Q3和Q4解決問題的例子。</p><p>　　S1 - 計劃情況 - 在情況S1中，教師寫下一個例子的文本，然后“投射”到他的解決方案中。學(xué)生會受教師意識的影響。這也是一種涉及學(xué)生活動的情況。學(xué)生可以通過構(gòu)建立方體截面的方式解決問題，然后發(fā)現(xiàn)其他點是否為平面點;或者學(xué)生解決他編寫平面的參數(shù)方程的問題，并且發(fā)現(xiàn)第四點是否是該平面的點;

98、或者他將向量的共線性或共面性應(yīng)用于“重心”問題的解決（但這種解決方案沒有任何假設(shè)）。</p><p>　　S0 - 教學(xué)情境 - 在這種情況下，我們分析并對新知識進行總結(jié)并給出結(jié)論以及制定問題。老師考慮到教學(xué)的目標(biāo)，并對學(xué)生的解決方案進行評價。這是一種分析教師工作和分析學(xué)生工作的情境，而教學(xué)情境將是教學(xué)過程的結(jié)果。</p><p><b>　　結(jié)論</b></

99、p><p>　　在教授立體幾何時出現(xiàn)的問題是數(shù)學(xué)立體幾何問題的某一部分。有了這個問題，也就解決了立體幾何問題。具體來說，學(xué)生有問題解決問題，學(xué)生可以參照他們在中學(xué)的知識水平和學(xué)習(xí)水平在不同的理論框架下解決問題，但學(xué)生只有用學(xué)習(xí)過的簡單算法才能解決問題。</p><p>　　研究的目標(biāo)是提出學(xué)生知識應(yīng)用在解決立體幾何中不同部分數(shù)學(xué)問題的知識方面存在的問題，并通過教學(xué)實驗驗證假設(shè)來驗證其有效性或無

100、效性。</p><p>　　實驗也在大學(xué)（大學(xué)生是未來的數(shù)學(xué)教師）中進行，并證實我們在應(yīng)用學(xué)生的知識中存在類似的問題，這些問題來自立體幾何學(xué)中不同部分的數(shù)學(xué)以及其他學(xué)科知識。因此，教師更喜歡教學(xué)中的具體部分（在數(shù)學(xué)課上），所以老師經(jīng)常摒棄具有統(tǒng)一性的例子。</p><p>　　這項工作的結(jié)論包括各個任務(wù)的集合，我們（當(dāng)然）希望從立體幾何中擴展任何有趣的應(yīng)用程序問題（任務(wù)），這可以提高數(shù)學(xué)課

101、程的質(zhì)量。 因此，作為中學(xué)數(shù)學(xué)書籍文本的一部分，對這些任務(wù)的收集并進行修正以供實際使用。 在接下來的實驗中，我們希望嘗試寫出教育性論文來解決立體幾何體的某些部分問題。 課本教材將特別針對立體幾何中各個部分知識之間的協(xié)調(diào)性，從而更多提及科目之間的關(guān)系，即提到應(yīng)用其他學(xué)科（尤其是物理學(xué)）的知識應(yīng)用于立體幾何教學(xué)和解決立體幾何問題的可能性。</p><p><b>　　參考文獻</b></p

102、><p>　　Balacheff N.: Une étude des processus de preures en mathématiques chez des éléves de college, Gre-</p><p>　　noble, 1998</p><p>　　Berthelot R.: The role of sp

103、atial knowings in the elementary teaching of geometry, Zborník brati-</p><p>　　slavského seminára z teórie vyu?ovania matematiky, 2001</p><p>　　Bereková H., Földesi

104、ová L., Hríbiková I., Regecová M., Tren?anský I.: Slovník teórie didak- tických situácií, 1. ?as?, Zborník 4 príspevkov na seminári z teórie vyu?ovani

105、a matematiky,</p><p>　　Univerzita Komenského, Bratislava, 2001</p><p>　　Bereková H., Földesiová L., Regecová M., Krem?árová L., Slávi?ková M., Tren?a

106、nský I., Vankú? P., Zámo?íková Z.: Slovník teórie didaktických situácií, 2. ?as?, Zborník 5 príspevkov na</p><p>　　seminári z teórie vyu?

107、ovania matematiky, Univerzita Komenského, Bratislava, 2003</p><p>　　Brousseau G.: Fondaments et méthods de la didactique des mathématiques, Recherches en Didac-</p><p>　　tique des