> q` bjbjqPqP 4H::@ppppL L L ` HoHoHoHop` qTyyyy$hL OőNOOppyy˻մմմOpVyL yմOմմL yqZHo̷0Է,L FYմ!.9OOOO` ` ` YcD` ` ` c` ` ` ppppppADVANCED PROBLEM SOLVING ACTIVITIES: how do STUDENTS navigate TERRA INCOGNITA? pros and cons
Oleksiy Yevdokimov
University of Southern Queensland, Australia
Despite wide research literature on problem solving the essence of thinking processes connected with attempts to approach and solve difficult mathematical problems is still not being developed to its maximum potential. The special focus of the paper is on what actually happens in classrooms or with individuals solving difficult problems, what theoretical conceptions can be elaborated from practice. The four directions are distinguished as the most important in students work on difficult problems. Brief descriptions and various patterns of student engagement with problems for each direction are provided.
1. Introduction
Very often advanced problem solving activities, or more exactly, mental processes connected with attempts to approach and solve difficult mathematical problems, are hidden from observation and resemble an iceberg, in that their major part remains invisible all the time. Probably such a situation isnt extraordinary since any extra difficulties in problem solving require extra efforts from students to work on. This high level cognitive demand gives a certain influence on students perception of their work and emphasises the role of intuition and Aha! moments. Moreover, the most talented students find their solutions so natural that any explanations are not necessary. In an interview conducted in 2004 by email, Dr Terence Tao reflected on his past experiences as a profoundly gifted student, in addition to his current experiences as a highly celebrated young mathematician (Muratory et al, 2006). Answering a question about his experience in problem solving and problem finding he commented:
It certainly is a pleasure to work something out in mathematics, even it is something that has been done before; it is a certain satisfaction in having click and you see how something works, or when a concept [that] could be explained only unclearly and in a complicated manner can now be done in a clean and transparent manner. A large part of the task of problemsolving in mathematics is simply to see things the right way, and to realize certain heuristics or insights, which are well worth learning in themselves and make you appreciate the field as a whole a lot better (ibid., p.312).
Does the answer above seem intriguing and unusual? Or, is this a standard way of highlevel thinking of a mathematically gifted person that few others can achieve? Probably both yes and no, are equally applicable to both questions. Surely many factors of mathematical cognition depend on individual psychological features. Focusing of interest and narrow specialisation on such features may be the best way to advance the frontiers of knowledge. However, analysis and evidence of how to see things the right way works is no less essential for understanding the art of advanced problem solving.
Occurring twenty years before Taos interview, another example is worthy of mention. This dialogue took place at the Science Museum in Paris, where Professor Serge Lang provided a public lecture to a Saturday afternoon audience (Lang, 1985). A brief extract follows:
Antoine, high school student. You told us that the formulas EMBED Equation.3 and EMBED Equation.3
give all the rational solutions of EMBED Equation.3 , except EMBED Equation.3 and EMBED Equation.3 . Can you give us the proof now?
Serge Lang. Yes, naturally, I had even hoped somebody would ask that question earlier. The proof is easy. Suppose that ( EMBED Equation.3 , EMBED Equation.3 ) is a rational solution. Let EMBED Equation.3 ; and dont ask me where it came from, with a little ingenuity you could discover it yourself. (ibid., p.60). EMBED Equation.3
If we compare the two quotations, we can easily see they are harmoniously and mutually complementary. Also, they seem clear and easy to follow. But, the ways to achieve that clarity are complex, sometimes not wellstructured and described as magical moments in mathematics classrooms (Barnes, 2002), the sense of Aha! in mathematicians practice (Burton, 1999) or based upon personal experience of outstanding mathematicians (Hadamard, 1945; Poincar, 1952). Both examples emphasise the importance of research on advanced problem solving, in particular: (i) on students perception of a difficult problem; (ii) on factors that have the most essential influence on students mathematical thinking and reasoning while dealing with a difficult problem; (iii) on students obstacles to finding the solution to such a problem. An account of all that, therefore, is the purpose of this paper, in as much detail and with as many illustrations as space will allow. The special focus is on what actually happens in classrooms or with individuals solving difficult problems, what theoretical directions we can elaborate from practice, and which practical implementations can be further developed on the theory basis.
2. TERRA INCOGNITA of ADVANCED PROBLEM SOLVING
Despite wide research literature on problem solving and recognised consensus on the importance of problem solving in the mathematics curriculum (Australian Education Council, 1991; NCTM, 1980, 1989, 2000) this area is still not being developed to its maximum potential. Taplin (1998) suggested that this has happened
because not enough is known about how people best acquire problemsolving skills or how they can best be taught them. A great deal has been written and debated about problem solving but it is still clear that there are difficulties associated with teaching people how to succeed at it. (p.146)
Furthermore, even less is known about the essence of thinking processes connected with attempts to approach and solve difficult mathematical problems, where students are not supposed to share the common belief that mathematics problems are always solved in less than 10 minutes, if they are solved at all (Schoenfeld, 1985b, p. 372).
In this section, we consider theoretical framework, methodology of the research and four directions that we distinguish as the most important in students work on difficult problems. Some of them are not new. However, we did not have the intention to develop innovative approaches and create a new teachinglearning environment for advanced problem solving activities. We simply analysed the current situation. We suggest that research on students abilities in advanced problem solving will focus, in particular, on: (i) the role of prior knowledge and using special techniques (special constructions); (ii) the role of multiple solutions that provide diversity of thinking strategies and vice versa, and cognitive benefits of the process; (iii) the development of skills to see the links between problems and their solutions and (iv) the structure of different kinds of insight and Aha! moments. Brief descriptions and various patterns of student engagement with problems for each direction are provided below. Whenever possible we were trying to support our analysis with the most appropriate reference in the literature, not necessarily the oldest one. Of course, the list of references is far from exhaustive.
2.1. Theoretical framework
Questions related to advanced problem solving formed a research base of the study that the author focused on for some years. Different aspects of the research are described in several papers (Yevdokimov, 2005a, 2005b, 2006, and 2007). This paper highlights the linkage between mathematical problems and student cognition. How do they depend on and influence each other? Why do some problems have more influence to stimulate cognitive aspects than others? Why are most gifted students so reluctant to uncover the details of how they approach difficult problems? Schoenfeld (1985a) suggested four factors which can interact to affect problemsolving performance. These are (i) the problem solvers mathematical knowledge; (ii) knowledge of heuristics; (iii) affective factors which influence the way the individual views problem solving and (iv) the managerial skills associated with selecting and implementing appropriate strategies. We note that these factors can be taken into account in relation to the problem posing process. The focus in this paper is on the relationships between the factors discussed by Schoenfeld, in particular, how managerial skills relate to the other factors. We suggest, not neglecting the role of other factors, that the fourth one seems to be of paramount importance in the case of mathematical contests. Berzsenyi (1999) notes that to solve a problem, one usually needs at least one bright idea or a nonroutine application of some method (p. 186). Indeed, most nonroutine problems, though not all, are posed by developing certain ideas that form the base and environment of a problem, and lead to the desired result. Silver and Mamona (1989) noticed that students often posed problems without any regard to how to solve them [in that study O.Y.]. Though, such a situation happens in mathematics research and nonroutine problem posing, we tried to avoid using such problems in the study. We suggest that the purpose of problem solvers is to identify these ideas and provide an adequate response to them, i.e. to find a solution without clear knowledge of what those ideas can be about at the start of problem solving. If they had such knowledge, then that problem would be a routine one. Therefore, we stay in the situation, where a routine task to one person might not be a routine to another (MamonaDowns & Downs, 2005, p. 385). A problem structure through interrelation of two processes is given in Figure 1 (small circles symbolise the concept of ideas used either in problem solving or problem posing, whereas arrows show problem solvers attempts to identify those ideas for solution, and problem posers intention to develop them into a problem respectively).
SHAPE \* MERGEFORMAT
Figure 1. A problem structure through interrelation of two processes
This structure shows the focus and scope of the research through the links between the two processes. However, it doesnt give methods to apply. To find out more of what is going on in the problem solvers mind we use the following conceptual framework shown in Figure 2. As perception of a problem by the student, or students perceptual understanding of a problem we define his/her awareness of that problem, i.e. any information pointed out by the student within a period of time for perusal and before actual start of problem solving activities. It can refer to mathematical knowledge related to the problem as well as to students expectations or anything else. Perception of solution is a broader conception that includes perception of a problem, students views on solution and their transformation during problem solving. Again, it includes mathematical, psychological and sociocultural aspects of individuals.
Figure 2. Advanced problem solving framework
2.2. Methodology
We did observation of problem solving activities, conducted interviews with current students, organised discussions with former students, who were involved in different mathematical contests. They are mostly professional mathematicians now, and experienced mathematics teachers. This methodology concentrates on students conceptual constructions and their cognitive demands. The main goal is to analyse the students constructions in problem solving and stimulate the students mental activity. We claim that all students we worked with were gifted in mathematics since they had been selected over different years as the top 20 best mathematics students in Australia (selection based on results of their performances in different contests) and, therefore, invited to the School of Excellence organised by the Australian Mathematics Trust, held in April and December each year.
Amongst the questions we discussed with students, mathematicians and teachers were the following: How much helpful information can the statement of a problem provide? What part of the problem solving process is the most difficult? What can be undertaken if nothing comes to mind with a particular problem? How are ideas sorted out, if you have few of them on the list simultaneously? What is the reason (in students opinion O.Y.) that an interesting approach to solution has been missed? How often did anyone get experience of a sudden flash of understanding about something he/she didnt follow? More specific questions had been asked with respect to particular problems. Some of them we used as a part of questionnaire, while others were answered discussing special constructions, multiple solutions and links between different problems. Observations, thinking strategies and patterns of students work are presented in sections 2.3 2.6.
2.3. Prior knowledge and using special techniques (special constructions)
The notions of knowledge and special techniques are considered together in many contexts of problem solving (we use a slightly different term special constructions for the case of geometry only). MamonaDowns and Downs (2005) describe such a situation as one of the levels of the efficiency of knowledge as a source for problem solving the formatting of the knowledge base in special ways that are especially suitable for applications (p. 394). Discussion of techniques is given in MamonaDowns and Downs (2004). From experimental point of view we found it is quite difficult to differentiate prior knowledge and special techniques in the case of advanced problem solving. A good example of a special technique in the context of culture and knowledge of mathematics is given in Gowers (1999). We suggest that the more development both notions receive on this level, the harder is to identify the difference between them. The two problems of the 48th International Mathematical Olympiad (IMO, 2007) held in Hanoi, Vietnam, are definitely the case to be considered in the theory of problem solving. This is, also, the case fostering further research on advanced problem solving theoretical perspectives from practice. Solution of Problem 5 is based on a special technique called the Vieta Jumping method or root flipping (Ge, 2007), while another special algebraic technique Combinatorial Nullstellensatz (Alon, 1999) is a key component for Problem 6 solution. Both problems illustrate diversity of views on the role of prior knowledge and using special techniques and how they affect thinking strategies.We dont give the detailed account of the students work on these problems since this is a matter of another paper. Instead, we provide an example of a quite simple nonroutine problem discussing the same questions.
Problem 2.3.1
If EMBED Equation.3 and EMBED Equation.3 , find EMBED Equation.3 .
Solution
We can consider EMBED Equation.3 as roots of a cubic equation. Then EMBED Equation.3 or EMBED Equation.3 , where EMBED Equation.3 . Substituting EMBED Equation.3 and EMBED Equation.3 into the equation we get EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 . Hence, EMBED Equation.3 .
Some comments on the problem and solution follow:
Student K: This is a nice method [technique], and Ill be on track using it with other problems.
Student D: This is not a technique, this is much deeper This is an idea that gives me understanding what I can do with the stuff around the problem what I can do in another problem, if I think it should be done there.
Student A: This idea is one of many others I keep in mind. I call them mustknow things. If I use some of them I can change [modify O.Y.] the problem to get it simpler. It works well! Though, sometimes I am a little bit stuck what to do next
In addition we sought students comments on the following: What do you think this problem is about? One of the answers was extremely interesting: Its about specific relationships between specific numbers. We note that this is more than just a characterisation of the problem. This is a brief and precise description of students thinking (in his own words) with respect to that particular problem. It is evidence of what is going on in the students mind and exactly one of the situations that give an opportunity for researchers to look behind the scenes in problem solving activities.
We observed that many students start problem solving with attempts to use knowledge of techniques the most familiar for them. After that, if a problem hasnt been straightforward [students expression O.Y.] and final outcome is still not achieved, the next stage of solution search has required new approaches, or at least new design of old ones. On this stage prior knowledge took predominant position over other factors and even subfactors like special techniques based on prior knowledge. The higher disbalance between the factors we observed, the more diversity in final outcome we got. We cant state that the more balanced cases of students work (in terms of the factors) definitely would guarantee the better outcome in problem solving. Also, we observed that students performance in advanced problem solving depends on the specific areas such as algebra, number theory, geometry and combinatorics. We see this evidence much more complex than just students preferences and level of expertise at the certain area.
2.4. Multiple solutions, different thinking strategies and cognitive benefits
In this section we discuss two different solutions given by two different students. The emphasis is on cognitive benefits of their work. Describing cognitive benefits of multiple solutions Silver et al (2005) suggest that getting aware of at least another approach helps students become more flexible when solving similar problems and offers them additional strategies for their mathematical tool bag (p.297). We note that some solutions (e.g. the second solution below) may lead to formation of conceptual understanding of much broader questions, even theoretical ones that lie far behind the scope of initial problem. We call this feature a conceptual weight of the solution. However, conceptual weights of multiple solutions are different. This means that cognitive benefits of multiple solutions are different. Therefore, thinking strategies that correspond to multiple solutions are responsible for more or less cognitive benefit of each solution through their conceptual weight.
There is a widespread misconception amongst high school students. Most of them think about quadratic equations as routine exercises. However, there are many nonroutine problems connected with different properties of a quadratic function. One of such problems with multiple solutions is presented below.
2.4.1. Problem
A teacher wrote the polynomial EMBED Equation.3 on the board. After that each student, one after another, either increased by EMBED Equation.3 or decreased by EMBED Equation.3 either the coefficient of EMBED Equation.3 or the constant term, but not both. Finally EMBED Equation.3 appeared on the board. One student is sure that a polynomial with both integer roots necessarily appeared in the process. Is she right or not?
First solution
Since we are increasing or decreasing the coefficient or the constant by EMBED Equation.3 at a time, there must be a point where the coefficient exceeds the constant by EMBED Equation.3 , i.e. where the coefficient is EMBED Equation.3 and the constant EMBED Equation.3 , for some EMBED Equation.3 : EMBED Equation.3 . Then the corresponding polynomial can be factorised as EMBED Equation.3 , and this has two integer roots.
Second solution
If we consider EMBED Equation.3 , we can see that its value at EMBED Equation.3 will be changed by EMBED Equation.3 every time the either the coefficient or the constant is increased or decreased by EMBED Equation.3 . Let EMBED Equation.3 , then EMBED Equation.3 and EMBED Equation.3 . Therefore, in the process at some point a polynomial, EMBED Equation.3 must appear on the board for which EMBED Equation.3 . In other words, EMBED Equation.3 is a root of EMBED Equation.3 , and the other root must be EMBED Equation.3 , by Vietas formula.
2.4.2. Analysis of the solutions
Both solutions form the case, where solutions themselves and their cognitive benefits provide satisfactory evidence of how students approached the problem, which thinking strategies they used, and what we can expect from them in solving similar problems. The second solution has heavy conceptual weight, which goes as far as to development of calculus concepts, intermediate value theorem, and other properties of continuous functions. The first solution is much lighter in the conceptual sense, it focuses on a specific model given in the problem, and may not have resources to be extended in other areas. The first thinking strategy, corresponding to the first solution, is constructed on the base of one mathematical object only. It is easy to follow, but difficult to modify and apply in other problems. Whereas, the second strategy refers to some mathematical objects, based on relationships between those objects rather then on objects themselves, and, due to this flexibility can be modified to other applications.
2.5. Links between problems and their solutions
Examples of students work and their analysis are omitted in the congress version of the paper due to the space restrictions. We followed Atiyahs views (1984) of understanding mathematics for modelling problem solving activities in this direction. We just note that the skill of recognising similarities amongst problems and links between them is one of the most influential factors in advanced problem solving activities. Analysis of its further development, teaching and learning aspects, and interrelations with other factors is important part of researching problem solving.
2.6. Two different kinds of insight
The extralogical processes of insight and illumination remain into the educational research limelight. Multiple cases of a sudden flash of understanding or the Aha! experience are described in the mathematics education literature (Davis and Hersh, 1980; Devlin, 2000; Krutetskii, 1976; Rota, 1997). Polya (1965) spoke of a sudden clarification that brings light, order, connection, and purpose to details which before appeared obscure, confused, scattered and illusive (p. 54). However, still now the nature of the sudden realisation of new knowledge (Barnes, 2002, p. 85) needs further specification, in particular in advanced problem solving. We distinguish two different kinds of insight we observed in students work on difficult problems.
2.6.1. Logical insight
This kind of insight is mostly grounded on using prior knowledge, but comes to mind instantly, like a flash of new understanding, without any immediate linkage to thinking strategies used before. However, afterwards a problem solver is able to restore and explain all logical chains in solution, including those ones related to insight. Or, at least that part of solution, which directly connected with insight can be clearly argued afterwards. Students enjoy such experience, though they exaggerate its importance. Appearances of this kind of insight occur quite often in advanced problem solving activities. The following example gives a pattern of students work. The part of solution connected with insight is shown in bold italics.
2.6.2. Problem (Australian Mathematical Olympiad, 1983)
EMBED Equation.3 is a triangle and EMBED Equation.3 is a point inside it, EMBED Equation.3 . The perpendiculars from EMBED Equation.3 to EMBED Equation.3 , EMBED Equation.3 meet these sides at EMBED Equation.3 , EMBED Equation.3 respectively, and EMBED Equation.3 is the midpoint of EMBED Equation.3 . Prove that EMBED Equation.3 .
Solution
Figure 3.
Let EMBED Equation.3 . Lets denote a midpoint of EMBED Equation.3 as EMBED Equation.3 , and a midpoint of EMBED Equation.3 as EMBED Equation.3 . EMBED Equation.3 is a rightangled triangle, we can construct a circumcircle for this triangle, where the point EMBED Equation.3 is a circumcentre and EMBED Equation.3 is its diameter. The most important thing in this construction is a fact that EMBED Equation.3 as a central angle and EMBED Equation.3 as radii. We can repeat the same construction with regard to EMBED Equation.3 . We get EMBED Equation.3 and EMBED Equation.3 . After that, since points EMBED Equation.3 and EMBED Equation.3 are midpoints for EMBED Equation.3 and EMBED Equation.3 respectively, we obtain EMBED Equation.3  EMBED Equation.3 , in particular EMBED Equation.3  EMBED Equation.3 . The same we apply for EMBED Equation.3 , where points EMBED Equation.3 and EMBED Equation.3 are midpoints for EMBED Equation.3 and EMBED Equation.3 respectively, and EMBED Equation.3  EMBED Equation.3 . Therefore, EMBED Equation.3 is a parallelogram.
So, we can conclude that EMBED Equation.3 . Hence, EMBED Equation.3 . The last step: If we consider triangles EMBED Equation.3 and EMBED Equation.3 , we get EMBED Equation.3 since EMBED Equation.3 and EMBED Equation.3 , EMBED Equation.3 since EMBED Equation.3 and EMBED Equation.3 . Thus, triangles EMBED Equation.3 and EMBED Equation.3 are equal, and EMBED Equation.3 .
2.6.3. Tricky insight
Another kind of insight can be described as related to the specific part of solution (a tricky place) that might or might not be identified by a problem solver, and not connected directly with other parts. But, getting through that specific part seems to be a necessary condition to complete problem solving successfully. Sometimes that specific part can be clearly argued afterwards, but in most cases it cant. This kind of insight happens much more rarely. Most cases we observed it was connected with managerial skills in the first instance and with other factors not so substantially involved (though, more evidence is required). Its appearance depends on complexity and structure of mathematical content used in a problem. Its application zone can vary from a simple illogical step in attempt to solve a problem to construction a complicated theory for some part(s) of solution. The second problem (Day 1) of the 36th United States of America Mathematical Olympiad (USAMO, 2007) is a good example and source for observation and illustration of this kind of insight.
3. further reflections
Schoenfeld (1985a) has noted a widespread belief that only the brightest students can succeed at problem solving (p.43). Hembree (1992) argued that this belief is not wellfounded. Our observation supports Hembrees conclusion. Saul (1999) suggested that the minds of highability students dont differ all that much from those of other students they are just more efficient (p. 83). Our observation supports this hypothesis. We noticed that gifted mathematics students, though not the brightest ones, after gaining more experience in problem solving, understand, for example, that there are few options for the first step in solving any problem. They can distinguish such situations, though not all students are able to explain their understanding clearly. Furthermore, our results show that insight may have a specific structure and, therefore, specific impact on students performance despite the fact that students couldnt always explain why they made a step in a certain direction.
Returning to the importance of prior knowledge in solving difficult problems, it is still unclear to what extent specific problems should be proposed in mathematical contests or, even in problems sections of mathematical journals. In the case of journals such a situation shouldnt be seen as disputable, but for any contest this question assumes some sense of controversy.
Characterising the level of difficulty of problems in Australian national contests Lausch (2007) pointed out the following problem as turned out to be hard (p.90):
Question 2 (AMOC Senior Contest, 2004)
Let EMBED Equation.3 be any nonnegative real numbers such that EMBED Equation.3 and EMBED Equation.3 . Prove that EMBED Equation.3 .
We talked with students about this problem and others with the indicated level of difficulty. We asked students to work out solutions, if a problem was unfamiliar for them. We observed that for most problems, which turned out to be hard, there was a great disbalance between students perception of a problem and his/her perception of solution. Also, we noticed that those students, who navigate through problem solving with different ideas, even unsuccessful ones, communicated more easily about their approaches than those, who applied an appropriate method (technique) and have got correct result quite quickly.
The answers to the questions raised and other ones from the context are extremely important for developing further theoretical framework of advanced problem solving as well as for practical implications in work with gifted mathematics students. Most highprofile students regularly participate in numerous mathematical competitions and for them to achieve best results their training should be grounded on a comprehensive theoretical basis.
References
36th United States of America Mathematical Olympiad (USAMO) (2007). http://www.maa.org/news/O7SOL.pdf
48th International Mathematical Olympiad (IMO) (2007). http://www.imo2007.edu.vn
Alon, N. (1999). Combinatorial Nullstellensatz, Combinatorics, Probability and Computing, 8, 729.
Atiyah, M. F. (1984). An interview with Michael Atiyah. Mathematical Intelligencer, 6, 919.
Australian Education Council. (1991). A National statement on mathematics for Australia schools. Melbourne: Curriculum Corporation.
Barnes, M. (2002). Magical moments in mathematics: Insights into the process of coming to know. In L. Haggarty (Ed.) Teaching Mathematics in Secondary Schools, 8398. Routledge/Falmer, London & New York.
Berzsenyi, G. (1999). Providing opportunities through competitions. In L. J. Sheffield (Ed.) Developing Mathematically Promising Students, 185190. Reston, VA: NCTM.
Burton, L. (1999). The practices of mathematicians: What do they tell us about coming to know mathematics? Educational Studies in Mathematics, 37(2), 121143.
Davis, P. J. & Hersh, (1980). The mathematical experience. Boston, MA: Birkhauser.
Devlin, K. (2000). The Maths Gene: Why everybody has it, but most people dont use it. London: Weidenfeld & Nicholson.
Ge, Y. (2007). The method of Vieta jumping, Mathematical reflections, 5, http://reflections.awesomemath.org/2007_5/vieta_jumping.pdf
Gowers, W. T. (2000). The two cultures of mathematics. In V. Arnold, M. Atiyah, P. Lax & B. Mazur (Eds.) Mathematics: Frontiers and Perspectives, 6578. American Mathematical Society.
Hadamard, J. (1945). The psychology of invention in the mathematical field. Princeton University Press.
Hembree, R. (1992). Experiments and relational studies in problem solving: A metaanalysis. Journal for Research in Mathematics Education, Vol. 23, 3, 242274.
Krutetskii, V. A. (1976). Psychology of mathematical abilities in schoolchildren. In J. Kilpatrick & I. Wirszup (Eds.), J. Teller (Transl.), Chicago and London: The University of Chicago Press.
Lang, S. (1985). The beauty of doing mathematics. Three public dialogues. SpringerVerlag.
Lausch, H. (2007). An Olympiad problem appeal. The Australian Mathematical Gazette, v.34, 2, 9092.
MamonaDowns, J., & Downs, M. L. N. (2004). Realization of techniques in problem solving: The construction of bijections for enumeration tasks. Educational Studies in Mathematics, 56, 235253.
MamonaDowns, J. & Downs, M. (2005). The identity of problem solving. Journal of Mathematical Behaviour, In Special Issue: Mathematical problem solving: What we know and where we are going, v.24, 385401.
Muratory, M. C., Stanley, J. C., Ng, L., Ng, J., Gross, M., U., M., Tao, T. & Tao, B. (2006). Insights from SMPYs greatest former child prodigies: Drs Terence (Terry) Tao and Lenhard (Lenny) Ng reflect on their talent development. Gifted Child Quarterly, Fall 2006, Vol.50, 4, 307324.
National Council of Teachers of Mathematics (1980). An agenda for action: Recommendations for school mathematics of the 1980s. Reston, VA: NCTM.
National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Poincar, H. (1952). Science and method. New York, NY: Dover Publications, Inc.
Polya, G. (1965). Mathematical discovery: On understanding, learning and teaching problem solving, New York: Wiley.
Rota, G. (1997). Indiscrete thoughts. Boston, MA: Birkhauser.
Saul, M. (1999). A community of scholars: Working with students of high ability in the high school. In L. J. Sheffield (Ed.) Developing Mathematically Promising Students, 8192. Reston, VA: NCTM.
Schoenfeld, A. H. (1985a). Mathematical problem solving. London: Academic Press.
Schoenfeld, A. H. (1985b). Metacognitive and epistemological issues in mathematical understanding. In E. A. Silver (Ed.) Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives, 361379. Hillsdale, NJ: Lawrence Erlbaum.
Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C. & Font Strawhun B. T. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behaviour, In Special Issue: Mathematical problem solving: What we know and where we are going, v.24, 287301.
Silver, E. A. & Mamona, J. (1989). Stimulating problem posing in mathematics instruction. In G. Blume & M. K. Heid (Eds.) Implementing new curriculum and evaluation standards, 17. University Park, PA: Pennsylvania Council of Teachers of Mathematics.
Taplin, M. (1998). Management of problemsolving strategies. In A. McIntosh & N. Ellerton (Eds.) Research in Mathematics Education: A Contemporary Perspective, 145163. Perth, Western Australia: MASTEC, Edith Cowan University.
Yevdokimov, O. (2005a). On development of students abilities in problem posing: A case of plane geometry. Proceedings of the 4th Mediterranean Conference on Mathematics Education, v.1, 255267, Palermo, Italy.
Yevdokimov, O. (2005b). About a constructivist approach for stimulating students thinking to produce conjectures and their proving in active learning of geometry. Proceedings of the 4th Congress of the European Society for Research in Mathematics Education (ERME), 469478.
Yevdokimov, O. (2006). Inquiry activities in a classroom: Extralogical processes of illumination vs logical process of deductive and inductive reasoning. A case study. In J. Novotna, H. Moraova, M. Kratka & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. Vol. 5, 441449. Prague, the Czech Republic: PME.
Yevdokimov, O. (2007). Using the history of mathematics for mentoring gifted students: Notes for teachers. Proceedings of the 21st biennial conference of the Australian Association of Mathematics Teachers, 267275, Hobart, Tasmania.
Dr Oleksiy Yevdokimov is a Lecturer in the Department of Mathematics and Computing at the University of Southern Queensland, Australia. He is the editor of the problemsolving section of the Australian Senior Mathematics Journal. Research interests include problem solving and proof, gifted education and history of mathematics.
Problem solving
Problem solving
Factors influencing both processes
Factors influencing both processes
Problem solving
Factors influencing both processes
to see things the right way
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
ncing both processes
Factors influencing both processes
Factors influencing both processes
Factors influencing both processes
Problem solving
Problem solving
Problem solving
Factors influencing both processes
Factors influencing both processes
Problem solving
Factors influencing both processes
to see things the right way
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
ncing both processes
Factors influencing both processes
Factors influencing both processes
Factors influencing both processes
Problem solving
Factors influencing both processes
to see things the right way
wito see things the right way
to see things the right Observation and analysis
Observation and analysis
Problem posing
Problem posing
s the right way
with a little ingenuity you could discover it yourself
ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
h processes
Factors influencing both processes
to see things the right way
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
ncing both processes
Factors influencing both processes
Factors influencing both processes
to see things the right way
wito see things the right way
to see things the right Observation and analysis
Observation and analysis
Problem posing
Problem posing
s the right way
with a little ingenuity you could discover it yourself
ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
h processes
Factors influencing both processes
to see things the right way
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
ncing both processes
Factors influencing both processes
to see things the right way
Observation and analysis
Observation and analysis
Observation and analysis
Problem posing
Problem posing
Problem posing
Problem posing
to see things the right Observation and analysis
Observation and analysis
Problem posing
Problem posing
s the right way
to see things the right way
Observation and analysis
Observation and analysis
Observation and analysis
Problem posing
Problem posing
Problem posing
Problem posing
to see things the right Observation and analysis
Observation and analysis
Problem posing
Problem posing
s the right way
with a little ingenuity you could discover it yourself
ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
with a little ingenuity you could discover it yourself
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Observation and analysis
Problem posing
Problem posing
Problem posing
Problem posing
Problem posing
Problem posing
Problem posing
Problem posing
Problem posing
Problem posing
Problem posing
Problem posing
Problem posing
Problem posing
Problem posing
Problem posing
Difficult problem
Its perception by the student
Solution of a problem
Its explanation by the student;
Perception of solution by the student
Problem solving
First step of a solution;
Information extracted from a problem;
Final step of a solution;
Time constraints (not necessary)
Schoenfelds factors affecting problem solving
problem solvers mathematical knowledge;
knowledge of heuristics;
affective factors which influence the way the individual views problem solving;
managerial skills associated with selecting and implementing appropriate strategies
PQ^_q}B y z
0
1
Q
ѿѰxxtlg h5h}h5hXhKh6h4YhhlRhhah1.h%Vh%VmH sH h%Vh%VmH sH h%VhEhEmH sH h1.hlh1.h1.h1.mH sH h1.mH sH h>hmH sH hmH sH h>*hMhAhh(_rQ! Gm$$2/Q//1353gd$a$gdgdgdgdgdgdagdgd
DEXYZ[dexyz{taTjH h@hEHU%jCuK
hCJUVaJnH tH jh@hEHU%j)uK
hCJUVaJnH tH jh@hEHU%jtK
hCJUVaJnH tH jmh@hEHU%jstK
hCJUVaJnH tH jh@hEHU%jFtK
hCJUVaJnH tH jhUh01DEFGHI\]^_}~l_Xh@hjh#hEHU%jbvK
hCJUVaJnH tH h4YhjWh#hEHU%jvK
hCJUVaJnH tH jb
h#hEHU%jvK
hCJUVaJnH tH jnh#hEHU%juvK
hCJUVaJnH tH jhUh h5h@h5!@KGmE !!!!"""""*""""@#A##~$$$$$$a&b&u&&&v*}***
+++++!,#,4,{hhh'mH sH hdmH sH hmH sH h7smH sH hH2h%hh>8mH sH h{h{h{hlhh.FhhmH sH hhS82h;.2.0/2/3/4/K/L/M/N/O/Q/d//!0&0'080;0l0m0p00112.3/30313436383:3=3?3@3E3G3t3344U7V777yyhGhXh6hXhXmH sH hXhmH sH hhmH sH h`Gh.9nh6]/hh>8h>8hOEh>8hBhmH sH j(hUjhUjhUmHnHsHuhmH sH hh3/536393:3>3?3A3B3C3D3E3G3t336::BBsBBCgdFgdFgdkgd$dH$a$gd/dgdgdXgdX$a$gd$
a$gdgd78899::::::;;G;Z;};;;;;;<<<<<<<<<==C=I=====>>6>}}ڻλwnwfnhRqnH tH h/dhRq0J
hRq0Jh/dh&0Jh&h&0Jh&nH tH h(h(0Jh/dh/d0Jh/dh(0Jh/dh0Jh/dh "0Jhh0J6
h0J
hC0J
h&0Jh/dhux0Jhuxhh'hH2hXhX(6><>_>b>>>>>>>>>>> ??C?J?T?n?????@$@+@,@@<@F@O@m@z@@@@@@6A?AAAAAAܨ̖ƖƐxth
hh_'
h0J
h*B0Jh&:
h&:0J
hR0J
h"&0J
h1C0J
h14B0J
h20J
hoj0J
h0
`0J
h_'0J
h0J
h0JhThT0JH*
h20J
hT0Jh/dh0Jh/dh "0J
hC0JAAABBBBBB B!B4B5B6B7BCCCDCWC泦擆zgZzzj$hTwhTwEHU%jK
hTwCJUVaJnH tH jhTwUhTwj!hFhTwEHU%jhk0JCJhkhk0JCJhDthkhbChkhCphkh0hkh
h[hkh*Bhkhu&hkh0
`hkhA&hkhkhk5hkhkh
Q0CDtDQEIFHLLLPRRSSUU@XaXf\\gd;gd;gdXgdN?NDNNNNNNNO6O^J^M^V^^^~zh.hK)hhXJhGGhGGh)5hGGh9hBh>hkfbhh;h;nH tH h^AnH tH hhqnH tH hTnH tH hnH tH ht1nH tH hFO3nH tH h/CnH tH h3nH tH hq*nH tH /^^^^^_8_q_w_}____`!`,````7a;aaaaaaaab bbbbbbbbbbcccccccccdd0dBdidddddddhahhQh<h<h<h'h8h]Zhv&h]xhhGh[OhThF\hQuh'hzhA<h`h9e6hh>h>hkfbh_2h@:\^_a bd%effffkymmqqu3wwxx{gdgdgd$a$gdagd]xgd$gdIRgd]xgdhGgdgdgd>ddd$e%e&e9e:e;efPfQfdfefffgfzf{fff泦擆sfS%jK
h$CJUVaJnH tH juh$h$EHU%jԝK
h$CJUVaJnH tH jsh$h$EHU%jK
h$CJUVaJnH tH jqh$h$EHU%jK
h$CJUVaJnH tH joh$h$EHU%jhK
h$CJUVaJnH tH h$jh$Ujmh$h$EHUfffffffffffffffffffffggggggg g3g³桖s]Pjh'hEHU+jK
h'hCJUVaJnH tH jph'hEHU+jK
h'hCJUVaJnH tH jh'hUh'hhahah$j{huehXUmHsHhIRjyh$hEHU%jK
hCJUVaJnH tH h$jh$Ujwh$h$EHU3g4g5g6gJgKg^g_g`gagegfgygzg{gg~gggggggg h
hʴʑnaK+jK
h'hCJUVaJnH tH jh'hEHU+j]eJ
h'hCJUVaJmHsHjh'hEHU+jsK
h'hCJUVaJnH tH jh'hEHU+jaK
h'hCJUVaJnH tH h'hjh'hUjh'hEHU+jMK
h'hCJUVaJnH tH
hhh#h$h7h8h9h:hLhhhhhhhhhhhhhh
ii!iʽcSjhGYlh56EHU1jK
hGYlh56CJUVaJnH tH jhGYlh56EHU1jfJ
hGYlh56CJUVaJmHsHjhGYlh56UhGYlh56jh'hEHU+jK
h'hCJUVaJnH tH h'hjh'hUjh'hEHU!i"i#i$ii.iAiBiCiDiIiJi]i^i_i`iaibili{iiiiʴʑzpzbpI1jK
hGYlhk*56CJUVaJnH tH jhGYlh 56UhGYlh 56hGYlh56jfh'hk*EHU+juK
h'hk*CJUVaJnH tH j#h'hEHU+j`hJ
h'hCJUVaJmHsHh'hjh'hUjh'hEHU+jgJ
h'hCJUVaJmHsHiiiiiiiiiiiiiiiiiiiii
jɿɿ}mɿTDjghGYlhk*56EHU1j$K
hGYlhk*56CJUVaJnH tH jqhGYlhk*56EHU1jK
hGYlhk*56CJUVaJnH tH jhGYlhk*56EHU1jK
hGYlhk*56CJUVaJnH tH hGYlhk*56jhGYlhk*56UhGYlh56jhGYlh 56UjhGYlhk*56EHU
jj!j"j#j$j&j'j:j;jhGYlhk*56EHUkkkkkk1k2k3k4kGkHk[k\k]k^k`kaktkukvkwkykkkíÊ}gZPBjhGYlhk*56UhGYlh56jh'hk*EHU+j֤K
h'hk*CJUVaJnH tH j)h'hk*EHU+jʤK
h'hk*CJUVaJnH tH j3h'hk*EHU+jK
h'hk*CJUVaJnH tH h'hk*h'hjh'hk*Uj=h'hk*EHU+jK
h'hk*CJUVaJnH tH kkkkkkkkkkkkkkkkkkkllllllll+l,l?lͿnaYjhk*UjDh,hEHU%jlJ
hCJUVaJmHsHjh,hEHU%jkJ
hCJUVaJmHsHjhUhk*hGYlhhGYlh56jhGYlhk*56UjhGYlhk*56EHU1jK
hGYlhk*56CJUVaJnH tH hGYlhk*56?l@lAlBlGlHl[l\l]l^lglhl{ll}l~llllllllllllllllϼϜo\Ojhk*hk*EHU%jK
hk*CJUVaJnH tH jhk*hk*EHU%jK
hk*CJUVaJnH tH jmhk*hk*EHU%jnK
hk*CJUVaJnH tH jshk*hk*EHU%j\K
hk*CJUVaJnH tH hk*hjhk*Ujyhk*hk*EHU%jOK
hk*CJUVaJnH tH llllllllllllm m
mm
mmm1m2m3m4m9mϼϜӅwmTDwj4h'hGYl56EHU1jK
h'hGYl56CJUVaJnH tH h'hk*56jh'hk*56Uh'h56jhk*hGYlEHU%jݥK
hGYlCJUVaJnH tH jhk*hGYlEHU%jǥK
hGYlCJUVaJnH tH hk*hjhk*Ujhk*hGYlEHU%jK
hGYlCJUVaJnH tH 9m:mMmNmOmPmZm`mamtmumvmwmym}m~mmmmmmm"n2n3n7nEnNn_ndnennnnn%oξuqmqieaqeqiqi]imimahoh8hYh"hE1hh]xh]xhIRh]xj)hGYlhGYlEHU%jK
hGYlCJUVaJnH tH hGYljhGYlUhh'h56j/h'hGYl56EHU1jK
h'hGYl56CJUVaJnH tH h'hGYl56jh'hGYl56U#%o(o/oooooopplpmp{pppppp
qq$q,q.q0q1qJq_qdqmqqqqqqr6sUsVs_s`ssuuuu!u.u0u7uuuv?v@v3wwwwwxĴĴĴĴļȰȠ
h`DPJhohPJhh_'h_'hQhMh.hzkghh
h0JhThZ0JH*
h2l0J
hZ0JhRhZoxhbhYBh8hx#E;xxxxxxxxxHxIxJx]x^x_x`xexfxyxzx{xxxxxxxxxxxxxn_YSYS
hoPJ
h8PJjhgKIhEHPJU%j6K
hCJUVaJnH tH jh`8hEHPJU%jQ*K
hCJUVaJnH tH jh`8hEHPJU%j)K
hCJUVaJnH tH jJhohEHPJU%j*K
hCJUVaJnH tH jhPJUhohPJ
hPJ xxEyIyyyyyyyzz{{}9}:}<}>}?}q}}}}}}}}&~@~q~~F־zrjhh6h/ch6hKBVhKBV\hKBV6]hKBVhojh0Jh2hH*hhUYh0JhUYhUY0JhTh0JH*
h0JhhhhDthDthh8h1ShLThpJ]ha
h.PJ
haPJ
h8PJ&:}}}K~~C62Rs4$7rgd1gdMjZgdgdgdgdFo'1BCЀҀԀ46IMRف2Â5̺̾̾̾~v~og`Xh5ph6hhhBh6h14Bh14BhXhX6hXh14Bh6]mH sH h6]hhhn\mH sH hn\hn\mH sH hn\hn\6hn\hh6]nH tH hhnH tH hnH tH h/ch6mH sH hmH sH hhh6"5RۃCńɄʄ΄ڄ>as*4zX$kÉ։ظذ؞؏xpxh;Cuh6h5phhh6hh6hhhu?h6h]Kh6h1h16]h1nH tH hehh6 h6h8]h6hMjZ6]hMjZhh6mH sH h{ h6mH sH hmH sH hh,`qrZb>FjpˌՌیˍ/ގ*ȏɏֲֺ֦֟֟֟֟֟֗thh6hhhh6hhu?h6h]Kh6h]Khh]h h6hh6h"nh6h/ch6mH sH hmH sH hhh$DmH sH h$D6]h$Dh0h06h0hNrtŋ/*
v_`abcdeϕЕѕҕӕԕgdH$gdgd!?gdgdgd$Dɏ
!"xΐސߐْDU[]ovDK]_`de$I\]fĺh0B*phh!?heh
h
h!?6h!?h!?h<h!?h!?hlh6H*]hlh6]hlhhhmH sH h6H*]h6]hmH sH hh6hhh.Εϕ֕ڕەߕ'():;HLMQRVW[~ږۖ*+,OPQRSZ^_cϗЗ˿hwdh0h5Ch0CJ$aJ0h0CJ$aJ0h5Ch0B*CJ$aJ0phh0B*CJ$aJ0phh0B*phh5Ch0h5Ch0B*CJaJ$phhqZh0h0hqZh0B*ph;ԕՕ֕וٕؕڕەܕݕޕߕ)*+;<=>?@ABCDEH$gdgdEFGHIJKLMNOPQRSTUVWXYZ[\ږۖgdۖ,QRSTUVWXYZ[\]^_`abcdgdЗї;<=Mgd/dgdz,wgd9:;LMZ^_cdhim'<=>abcdelpquЙљ $%)*./3489=>BCGHLMQtuvhwdh0hqZh0h5Ch0CJ$aJ0h0CJ$aJ0h0h5Ch0TMNOPQRSTUVWXYZ[\]^_`abcdefghijgd/djklmn'(>?cdefghijklmnopqgd/dqrstuvљҙәԙՙ֙יؙٙڙۙܙݙޙߙgdgd/dgd/dgdz,wgd
gdgd/d !"#$%&'()*+,./0123456gd/dgdz,wgd6789:;<=>?@ABCDEFGHIJKLMNOPQRgd/dRuv'(9:tuڛۛPQ^_H$gdgdԚ&'(9:HuڛۛP\]^ݜޜ./RSTwxyם
)*+<=KxݞޞS_`a0h5Ch0CJ$aJ0h0CJ$aJ0hqZh0B*phh0B*phhqZh0h0hwdh0O_ݜޜ/0TUxy
*+<=gd/dgdz,wgd=wxݞޞSTab23WXwxgdgd/d012UVWXkvwĠŠԠՠ%=>WXghwxyáġݡޡ'(78>Xpq̢#$^_ԣգ:;uvhqZh0hwdh0h0h5Ch0CJ$aJ0h0CJ$aJ0VŠƠՠ֠
>?XYhixygd/dgdz,wgdġšޡߡ ()89:;<=>qrgd/d#$^_ԣգ:;uv&'QRǥgdz,wgdgd/d'QRǥȥ>hiަߦ23LMfgΧϧ67PQjkȨɨب٨ ()89HIXYhixy̩hh0CJaJhqZh0h0hwdh0Vǥȥ=>hiަߦ34MNghϧЧgdgd/d78QRklɨʨ٨ڨ
gdgd/dgdz,w)*9:IJYZijyz̩$
hd8^`a$gdgd/dgdz,wgd"@AHZgstuժh!?hehz,wh0hh0CJaJh0CJaJ#IJZtժ֪.Gtt$
hd8`a$gd$
hd8^`a$gdgd/d
hLd8^`Lgd
hd8`gd
hd8^`gdgdgd/d
hd8^`gd
gd!?gd/dgd/d$
hLd8^`La$gd21h:pz,w. A!n"n#$%mDd
b
c$A??3"`?2gvJG>}H{̾Di`!gvJG>}H{̾@ 8 dYxcdd``Ng2
ĜL0##0KQ*
Wä&d3H1)fY+A<6@=P5<%!@5@_L ĺE1X@3V;3)V ZZpAdeĥl
jT 31Ag>2sC='B0=Lp{&L(ӏ_W`ܤ\
+_3m`.l+1N`eENOLLJ% {:@ćOBaJbDd
lb
c$A??3"`?2#8Cg~nUAuEi`!#8Cg~nUAuE@8 Nxcdd``g2
ĜL0##0KQ*
WفMRcgbR vxl3zjxK2B*Rj8 :@u!f010mc wfRQ9A $37X/\!(?71a}![K$0 Adeɓըb^?c˘@FVByH!0Y2nO5
?)į?&I9eӄj$T}b[Y.p샃b;^+KRs@0u3t5#hw'KDd
thb
c$A??3"`?2a@ Jjxqi`!ia@ Jjxr @
7xcdd``dd``baV d,FYzP1n:&&v! KA?H1
ㆪaM,,He` @201d++&1b@CNl:]j mĵю
հbcd~adaT K3BX'27)?Aꃴ$NwCa߀dm%kcH00LWTrAc`ƍ;LLJ% {: 2eq.Dd
b
c$A??3"`?2x&&%v_IT^i`!L&&%v_I@Hxcdd`` @c112BYL%bpu ; dn%[{It7F=> **Ipp@H{F&&\ s:@g!t?1: \&Dd
D@b
c$A??3"`?2p8"q؎L i`!D8"q؎ xcdd`` @c112BYL%bpu<rԩi`!<rԩ:@`!xcdd`` @c112BYL%bpu1@`2`30Dd
b
c$A??3"`?2?+ǃXZg
i`!+ǃXZg:`!xcdd`` @c112BYL%bpu5ORDd
plb
c$A??3"`?
!"#$%&'()*+,./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{}~#
Root Entry F@ZData
aWordDocument4HObjectPool
ۤZ@Z_1270576198FۤZۤZOle
CompObjfObjInfo#(+.369>ADGJMNQTUX[^chknqtwz}~
FMicrosoft Equation 3.0DS EquationEquation.39q7M1<2/
x=1"t2
1+t2
FMicrosoft Equation 3.0DS EqEquation Native i_1270576243FۤZۤZOle
CompObj
fuationEquation.39q7;@?.<2,
y=2t1+t2
FMicrosoft Equation 3.0DS EquationEquation.39qObjInfo
Equation Native
W_1270576356FۤZۤZOle
CompObj
fObjInfoEquation Native U_1270576425XFۤZۤZ79<2
x2
+y2
=1
FMicrosoft Equation 3.0DS EquationEquation.39q7(r5
x="1Ole
CompObjfObjInfoEquation Native 5_1270576451,FۤZۤZOle
CompObjfObjInfo
FMicrosoft Equation 3.0DS EquationEquation.39q7s7
y=0
FMicrosoft Equation 3.0DS EquationEquation.39qEquation Native 1_1270576757"FۤZۤZOle
CompObj fObjInfo!Equation Native )_1270576774$FۤZۤZOle
!7
05,
x
FMicrosoft Equation 3.0DS EquationEquation.39q7
x07,
y
FMicrosoft Equation 3.0DS EqCompObj#%"fObjInfo&$Equation Native %)_1270576821@)FۤZۤZOle
&CompObj(*'fObjInfo+)Equation Native *FuationEquation.39q7*h06,
t=yx+1
FMicrosoft Equation 3.0DS EquationEquation.39q_1270576738.FۤZۤZOle
,CompObj/fObjInfo0/Equation Native 0#_12709882133FۤZۤZOle
1CompObj242f7.<2,
FMicrosoft Equation 3.0DS EquationEquation.39q%4L
a+b+c=0ObjInfo54Equation Native 5A_1270988291;8FۤZۤZOle
7CompObj798fObjInfo::Equation Native ;9_1270988323=FۤZLZ
FMicrosoft Equation 3.0DS EquationEquation.39q[
abc=4
FMicrosoft Equation 3.0DS EquationEquation.39qOle
<CompObj<>=fObjInfo??Equation Native @bFD
a3
+b3
+c3
FMicrosoft Equation 3.0DS EquationEquation.39q%'D<
a,b,c_12709886276OBFLZLZOle
BCompObjACCfObjInfoDEEquation Native FA_1270988694GFLZLZOle
HCompObjFHIf
FMicrosoft Equation 3.0DS EquationEquation.39qj
x"a()x"b()x"c()=0ObjInfoIKEquation Native L_1270988933LFLZLZOle
OCompObjKMPfObjInfoNREquation Native S_1270988860EJQFLZLZ
FMicrosoft Equation 3.0DS EquationEquation.39qH~
x3
"a+b+c()x2
+Ax"abc=0
FMicrosoft Equation 3.0DS EqOle
VCompObjPRWfObjInfoSYEquation Native ZMuationEquation.39q1D
A=ab+bc+ca
FMicrosoft Equation 3.0DS EquationEquation.39q_1270989043'wVFLZLZOle
\CompObjUW]fObjInfoX_Equation Native `5_1270989067[FLZLZOle
aCompObjZ\bf?L
a,b
FMicrosoft Equation 3.0DS EquationEquation.39q
$$
c
FMicrosoft Equation 3.0DS EqObjInfo]dEquation Native e)_1270989172Yc`FLZLZOle
fCompObj_agfObjInfobiEquation Native jK_1270989210eFLZLZuationEquation.39q/x,4L
a3
=4"Aa
FMicrosoft Equation 3.0DS EquationEquation.39qOle
lCompObjdfmfObjInfogoEquation Native pK/r
b3
=4"Ab
FMicrosoft Equation 3.0DS EquationEquation.39q/
c3
=4"Ac_1270989247^rjFLZLZOle
rCompObjiksfObjInfoluEquation Native vK_1270989307oFLZLZOle
xCompObjnpyf
FMicrosoft Equation 3.0DS EquationEquation.39qhx\}
a3
+b3
+c3
=12"Aa+b+c()=12ObjInfoq{Equation Native _1271008663mtFLZLZOle
CompObjsufObjInfovEquation Native S_1271008768hyFLZLZ
FMicrosoft Equation 3.0DS EquationEquation.39qu7%<2#
x2
+10x+20
FMicrosoft Equation 3.0DS EquationEquation.39qOle
CompObjxzfObjInfo{Equation Native )u
!+5&
1
FMicrosoft Equation 3.0DS EquationEquation.39qu
"+7&
1_1271008781~FLZLZOle
CompObj}fObjInfoEquation Native )_1271008753FLZLZOle
CompObjf
FMicrosoft Equation 3.0DS EquationEquation.39qu
(<2&
x
FMicrosoft Equation 3.0DS EquationEquation.39qObjInfoEquation Native )_1271008823FLZLZOle
CompObjfObjInfoEquation Native S_1271011946FLZLZu7!(6&
x2
+20x+10
FMicrosoft Equation 3.0DS EquationEquation.39qu
*5&
1Ole
CompObjfObjInfoEquation Native )_1271011906FLZLZOle
CompObjfObjInfo
FMicrosoft Equation 3.0DS EquationEquation.39qu
(<2&
1
FMicrosoft Equation 3.0DS EquationEquation.39qEquation Native )_1271011988FLZLZOle
CompObjfObjInfoEquation Native 1_1271012055FLZLZOle
u(*6&
n+1
FMicrosoft Equation 3.0DS EquationEquation.39qu
"(7&
n
FMicrosoft Equation 3.0DS EqCompObjfObjInfoEquation Native )_1271012070FLZLZOle
CompObjfObjInfoEquation Native )uationEquation.39qu
*7&
n
FMicrosoft Equation 3.0DS EquationEquation.39qu!*<2&
9d"nd"19_1271012115TFLZLZOle
CompObjfObjInfoEquation Native =_1271012230FLZLZOle
CompObjf
FMicrosoft Equation 3.0DS EquationEquation.39quC0<2.
x+1()x+n()
FMicrosoft Equation 3.0DS EqObjInfoEquation Native __1271014759FLZLZOle
CompObjfObjInfoEquation Native _1271014875FLZLZuationEquation.39qu`<2
f1
x()=x2
+10x+20
FMicrosoft Equation 3.0DS EquationEquation.39qOle
CompObjfObjInfoEquation Native 5u(<2&
x="1
FMicrosoft Equation 3.0DS EquationEquation.39qu
@*5&
1
FMicrosoft Equation 3.0DS Eq_1271014928FLZLZOle
CompObjfObjInfoEquation Native )_1271014966FLZLZOle
CompObjfuationEquation.39qu
*6&
1
FMicrosoft Equation 3.0DS EquationEquation.39qu`(*4&
f2
xObjInfoEquation Native )_1271014992FLZLZOle
CompObjfObjInfoEquation Native _1271015097FLZLZ()=x2
+20x+10
FMicrosoft Equation 3.0DS EquationEquation.39qu>*8&
f1
"1()=11Ole
CompObjfObjInfoEquation Native Z
u !["\$%&'()*+,./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ^]_a`bdcegfhjikmlnoprqsvtwyxz{}~_1271015155FLZLZOle
CompObjfObjInfo
FMicrosoft Equation 3.0DS EquationEquation.39qu>*6&
f2
"1()="9
FMicrosoft Equation 3.0DS EquationEquation.39qEquation Native Z_1271015258FLZLZOle
CompObjfObjInfoEquation Native f_1271015347FLZLZOle
uJ1<2/
gx()=x2
+px+q
FMicrosoft Equation 3.0DS EquationEquation.39qu,P35/
g"1()CompObjfObjInfoEquation Native H_1271015441FLZLZ=0
FMicrosoft Equation 3.0DS EquationEquation.39qu 3<21
"1
FMicrosoft Equation 3.0DS EqOle
CompObjfObjInfoEquation Native _1271015530FLZLZOle
CompObjfObjInfouationEquation.39qu 551
gx()
FMicrosoft Equation 3.0DS EquationEquation.39quP561
"qEquation Native <_1271015570FLZLZOle
CompObjfObjInfoEquation Native _1271110684FLZLZOle
!&+05:=@EJOTY^cfinsvy~
FMicrosoft Equation 3.0DS EquationEquation.39q/<2.
ABC
FMicrosoft Equation 3.0DS EquationEquation.39qCompObjfObjInfoEquation Native 1_1271110726FLZLZOle
CompObjfObjInfoEquation Native )
25.
P
FMicrosoft Equation 3.0DS EquationEquation.39q506.
"PAC= "PBC_1271110863NFLZLZOle
CompObjfObjInfo
Equation Native Q_1271110952FLZLZOle
CompObjf
FMicrosoft Equation 3.0DS EquationEquation.39q
0<2/
P
FMicrosoft Equation 3.0DS EquationEquation.39qObjInfoEquation Native )_1271110972FLZLZOle
CompObjfObjInfoEquation Native _1271111016FLZLZ35/
BC
FMicrosoft Equation 3.0DS EquationEquation.39qp37/
CAOle
CompObjfObjInfoEquation Native _1271111060
FLZLZOle
CompObj fObjInfo"
FMicrosoft Equation 3.0DS EquationEquation.39q
@36/
L
FMicrosoft Equation 3.0DS EquationEquation.39qEquation Native #)_1271111078FLZLZOle
$CompObj%fObjInfo'Equation Native ()_1271111124
FLZLZOle
)
37/
M
FMicrosoft Equation 3.0DS EquationEquation.39q
x18/
D
FMicrosoft Equation 3.0DS EqCompObj*fObjInfo,Equation Native )_1271111142FLZLZOle
.CompObj/fObjInfo1Equation Native 2uationEquation.39q38/
AB
FMicrosoft Equation 3.0DS EquationEquation.39q(35/
DL=DM_12711111760FLZLZOle
3CompObj4fObjInfo 6Equation Native 79_1271112099#FLZLZOle
8CompObj"$9f2ڨZ"]]u:G4xi`!pڨZ"]]u:G4F>xڕQJAͩw<.
"A3D{B&"&'
^"hZ,,}<)z잇S0737Ẽ˥ODBJԠ
fw9HjǦ`R!&nk2Nj䨡O#߾U)p&=TbȜ4k4qGntVĭ6OoQah=d{W*)tMGWA[3yխGf/%+M
7~?9\86Ls+ }ly.0n;HP$!\`6\v?;^Dd
Tb
c$A ??3"`?24yS i`!4yS ȽXJkxcdd``d!0
ĜL0##0KQ*
Wä2AA?H1Zc@øjxK2B*R\``0I3ڐDd9!D
3@@"?@Dd
b
c$A
??3"`?
2谷k?gfi`!^谷k?g*xH,xcdd``ed``baV d,FYzP1n:,56~) @ 730
ㆪaM,,He`0 @201d++&1t\>+sUsi#3l XcBܤl\a#d"V0~3
1B2p{LHۓ'V ~b%4T!>h
0y{qĤ\Y\
CD,Ā`+>fo+Dd
b
c$A??3"`?2u0GQpVQ<i`!I0GQpV8Hxcdd`` @c112BYL%bpu15bs=aF\_@Z+ac̨.Fb_?}$?]dBr% O@D{AAI9\>
'^FX@FhysAC;wLLJ%O@0u(2t5=m?Dd
D@b
c$A
??3"`?
2`Ug_7<i`!4Ug_7 xcdd``~ @c112BYL%bpucd۫"`k~``ÐI)$5dP"CDHg!_!v320eQDd
Tb
c$A??3"`?2WwRzZDi`!WwRzZDz` GXJxcdd``eb``baV d,FYzP1n:&>! KA?H1Z
ǀqC0&dT201
@2@penR~8ʅB+?Ob凸Usi#:FcVJBysm#dFfDS;!25c!ڽHq"vݫF{9vbRކsXd++&\^Ewuo3J'aEdHsLH۳?p6? ?* O2@习ͣ`pgdbR
,.Ie8r'P"CXY=,Dd
Whb
c$A??3"`?2!@I;Lu)ӻAdi`!!@I;Lu)ӻAҴ@xڝK@]?VZECt:?BmZ\tpU܄fm(M8~\H@! xؐQBDDeY/ߕ,ߏ{FŢ?
nuhh#Dx"5t$TdL;ֿwʿObte9r[F^rc0?'s?$,YI]:eKg*F\ 5`I0Xn>I32H0ѩcP0y{aĤ\Y\2C/2KDd
b
c$A??3"`?2>тU@Feq{'S&i`!тU@Feq{': `Ƚ!xcdd`` @c112BYL%bpu
]͙hG(i`!`E>
]͙b`hn .xcdd``6dd``baV d,FYzP1n:&v! KA?H1
ہqC0&dT20 `[YB2sSRs:V~.
_˙8ҔYt9Va+I惼I9Xa>#d]=.fIGd \䙋/\`'rAcc`cq!1#RpeqIj.2\E.B~`oq&ADd
@b
c$A??3"`?2ͣmkxGLKYg*i`!_ͣmkxGLKYb@0= xcdd``6dd``baV d,FYzP1n:&&v! KA?H130l
UXRY`7S?&meabMVKWMcX"+ONf*F\XANrL@&%%
W&00spE/(F p{]ln3
3_~8\Pp321)W2TePdh,ĀfhuBDd
t@b
c$A??3"`?2I)H\T${h,i`!`I)H\T${b
.xcdd``6dd``baV d,FYzP1n:&&v! KA?H1
ㆪaM,,He`0 01d++&1t\>H{TT0z0LJNF75 Ma`XdG!32H𰀌Ѕsb[l=p{XAF@UO.\Xh\0y{qĤ\Y\d.P"CDHg!t?0eK{Dd
p
hb
c$A??3"`?2M($u/i`!M($u@S!xcdd`` @c112BYL%bpu13XioJDd
<@b
c$A??3"`?2V}\ӑMTp1i`!hV}\ӑMTr` 6xcdd``dd``baV d,FYzP1n:&! KA?H1
ہqC0&dT20 KXB2sSRsbWv @:P.P5p#@N221g>#78\ dnpenR~CV
28
1*VpbaX@&TaAp0AFsF&&\u=@]
> 1,
pDd
b
c$A??3"`?2?tJH>4i`!tJH>:Yxcdd`` @c112BYL%bpu1@`2`7~Dd
b
c$A??3"`?2?:XUjkSHlg5i`!:XUjkSHlg:Yxcdd`` @c112BYL%bpuf8NO^Y7i`!f8NO^Y:@`!xcdd`` @c112BYL%bpu1@`2`3JDd
<@b
c$A??3"`?2;#YE9Дp9i`!h;#YE9Дr` 6xcdd``dd``baV d,FYzP1n:&! KA?H1
ہqC0&dT20 KXB2sSRsbWv DZP.P5p#@N24A#
3D.0sQ2727)?LU#C!2ݿd6!t{)r, *0@\ 8q#9#RpeqIj.Cw.]rDd
b
c$A??3"`?2?TwFN7uK<i`!TwFN7uK:Yxcdd`` @c112BYL%bpui`!i#]`D{{:Yxcdd`` @c112BYL%bputn:m"1@`2`P3KDd
b
!
c$A ??3"`? 2>'k"XŸ!0Di`!'k"XŸ!:@`!xcdd`` @c112BYL%bpuEi0D@db=>=$b#ܞ& y\P,8xa&#RpeqIj.w@]`
k[VyDd
Tb
#
c$A"??3"`?"2,OjŚ{]%WHi`!,OjŚ{]%z eXJexcdd`` @c112BYL%bpu#RpeqIj.\w.)Dd
hb
$
c$A#??3"`?#2;qqB>3^يJi`!;qqB>3^يb@vxcdd`` @c112BYL%bpu **Ipp@H{F&&\ s:@ćOB~bP^Dd
b
&
c$A%??3"`?%2?Цɪ~Sq,BOi`!Цɪ~Sq,B:Yxcdd`` @c112BYL%bpu1@`2`7Dd
b
'
c$A&??3"`?&2@v].hT}Qi`!v].hT:Yxcdd`` @c112BYL%bpu 1^Dd
Tb
,
c$A+??3"`?+2Dj;<8\e]i`!Dj;<8\ ~ XJJxcdd``ed``baV d,FYzP1n:&&6! KA?H1Zʎ
ㆪaM,,He`c&,ebabMa(Z`dAA^فòɴ,@@@ڈQ^JB; @c112BYL%bpu1l]Dd
b
/
c$A.??3"`?.2bJOdr)^}>di`!6JOdr)^}`0xcdd``> @c112BYL%bpudKF&&\W@]
? 1,VCDd
0b
4
c$A3??3"`?/2D斍QSŘ@m fi`!斍QSŘ@m<kHxcdd`` @c112BYL%bpu6Dd
@b
6
c$A5??3"`?123#MGt\ji`!T3#MGt
F "xcdd``~ @c112BYL%bpu1q#3XLDd
b
7
c$A6??3"`?22;OG{Rli`!OG{:Rx=N=a}3f+FB*$BXX$$*8BNAkv,_27!d%2CL! թehGJ2"S拠TSfRوCoؚ[OASspw<{rKT[ǻLޫ8kŅWrZzØPrQ~3Lt:\N 6*oi&8Dd
b
8
c$A7??3"`?32@xlٖ뵇!>eCni`!xlٖ뵇!>e:hHxcdd`` @c112BYL%bpu\P[.a
Cv;#RpeqIj.<w.NB;3X?9Dd
@b
;
c$A:??3"`?62?2
"k=^($ti`!2
"k=^(: xcdd`` @c112BYL%bpuxi`!"0yZs>:hxcdd`` @c112BYL%bpu
c$A=??3"`?92k㫝Ԃl,]ouGzi`!?㫝Ԃl,]ouHD
xcdd`` @c112BYL%bpuk +ssʏs]a>#d&>\Pp}6b;FLLJ%9 s: @fbb#3XJDd
3ga0
i
#Ah:"Ѕ.S܉nV`%i@=Ѕ.S܉nV`0&9#xՙLUeǟs!*+40F0%CCUA,H%0
*RֈjjmZжsd(X}y^z=n==ysϹ絈miX:E"_'(ȢD[&mR`/8
Tn¹IrcXoUQ=O~zꩆT1FxG\Òa 2OAX߂Xd1R.isɓŸ:+~iR[mG:F92>aٰak`%s*i?eM^5NWN
d;ʏD'k.ױN^5&t&Y<9
aLA9љd1v7NY#:U~FD:qg&:ĝu^d6
S
q9Fe9\Og:ӔNpoBgpߥnB瘦@ܕNZ>m=#YN'dD^_t?,M:'^C/{@h;hVs(m;`]7\QDg\ZwQoGSBkO=F:TϤR͖fPe0gjx;oY:̔}M,̌;VkJ
=8/MWrDt>'#҆ңf;re48w.Z=rYjqZ;}qR]hpuyl\_`pyvva^~ իs
FMicrosoft Equation 3.0DS EquationEquation.39q5&<2%
"PAC= "PBC=
FMicrosoft Equation 3.0DS EqObjInfo%;Equation Native <Q_1271112196!+(FLZLZOle
>CompObj')?fObjInfo*AEquation Native B_1271112269FLZLZuationEquation.39qi<2h
AP
FMicrosoft Equation 3.0DS EquationEquation.39q
)<2(
EOle
CCompObj,.DfObjInfo/FEquation Native G)_1271112289&?2FLZLZOle
HCompObj13IfObjInfo4K
FMicrosoft Equation 3.0DS EquationEquation.39q,5(
BP
FMicrosoft Equation 3.0DS EquationEquation.39qEquation Native L_12711123077FLZLZOle
MCompObj68NfObjInfo9PEquation Native Q)_1255368029<FLZLZOle
R
H,7(
F
FMicrosoft Equation 3.0DS EquationEquation.39q(
AMPCompObj;=SfObjInfo>UEquation Native V5_12711123525DAFLZLZOle
WCompObj@BXfObjInfoCZEquation Native [)
FMicrosoft Equation 3.0DS EquationEquation.39q
`,6(
E
FMicrosoft Equation 3.0DS EquationEquation.39q_1271112370FFLZLZOle
\CompObjEG]fObjInfoH_Equation Native `_1255368378:SKFLZLZOle
aCompObjJLbf,8(
AP
FMicrosoft Equation 3.0DS EquationEquation.39q%4
"MEP=2ObjInfoMdEquation Native eA_1271112446PFLZLZOle
gCompObjOQhfObjInfoRjEquation Native k9_1255368696UFLZLZ
FMicrosoft Equation 3.0DS EquationEquation.39q<2
ME=EP
FMicrosoft Equation 3.0DS EquationEquation.39qOle
lCompObjTVmfObjInfoWoEquation Native p5
"PBC
FMicrosoft Equation 3.0DS EquationEquation.39q%D
"LFP=2_1255368800I ZFLZLZOle
qCompObjY[rfObjInfo\tEquation Native uA_1271112565_FLZLZOle
wCompObj^`xf
FMicrosoft Equation 3.0DS EquationEquation.39q2<20
LF=FP
FMicrosoft Equation 3.0DS EquationEquation.39qObjInfoazEquation Native {9_1271112637]gdFLZLZOle
CompObjce}fObjInfofEquation Native )_1271112711iFZZ
H260
D
FMicrosoft Equation 3.0DS EquationEquation.39q
<2,
E
FMicrosoft Equation 3.0DS EqOle
CompObjhjfObjInfokEquation Native )_1271112727b{nFZZOle
CompObjmofObjInfopuationEquation.39q05,
AB
FMicrosoft Equation 3.0DS EquationEquation.39q07,
APEquation Native _1271112740sFZZOle
CompObjrtfObjInfouEquation Native _1255369198xFZZOle
FMicrosoft Equation 3.0DS EquationEquation.39qC
DE
FMicrosoft Equation 3.0DS EquationEquation.39qCompObjwyfObjInfozEquation Native _1271112758q}FZZOle
CompObj~fObjInfoEquation Native @08,
PB
FMicrosoft Equation 3.0DS EquationEquation.39q06,
DE_1271112784FZZOle
CompObjfObjInfoEquation Native _1271112803lFZZOle
CompObjf
FMicrosoft Equation 3.0DS EquationEquation.39q 1
PF
FMicrosoft Equation 3.0DS EquationEquation.39qObjInfoEquation Native _1255369368vFZZOle
CompObjfObjInfoEquation Native 5_1271112853FZZD
APB
FMicrosoft Equation 3.0DS EquationEquation.39q
3,
D
FMicrosoft Equation 3.0DS EqOle
CompObjfObjInfoEquation Native )_1271112866FZZOle
CompObjfObjInfouationEquation.39q
07,
F
FMicrosoft Equation 3.0DS EquationEquation.39q8,
ABEquation Native )_1271112881FZZOle
CompObjfObjInfoEquation Native _1271112892FZZOle
FMicrosoft Equation 3.0DS EquationEquation.39q05,
PB
FMicrosoft Equation 3.0DS EquationEquation.39qCompObjfObjInfoEquation Native _1271112906FZZOle
CompObjfObjInfoEquation Native 0 1
DF
FMicrosoft Equation 3.0DS EquationEquation.39qX3,
EP_1271112918FZZOle
CompObjfObjInfoEquation Native _1271112947FZZOle
CompObjf
FMicrosoft Equation 3.0DS EquationEquation.39qH0!1
DEPF
FMicrosoft Equation 3.0DS EquationEquation.39qObjInfoEquation Native 5_1255369645FZZOle
CompObjfObjInfoEquation Native I_1255369751FZZD
"DEP= "PFD
FMicrosoft Equation 3.0DS EquationEquation.39q0d
"MED= "LFDOle
CompObjfObjInfoEquation Native I_1271113039FZZOle
CompObjfObjInfo
FMicrosoft Equation 3.0DS EquationEquation.39q)<2(
MED
FMicrosoft Equation 3.0DS EquationEquation.39qEquation Native 1_1271113052FZZOle
CompObjfObjInfoEquation Native 1_1271113070FZZOle
@,5(
LFD
FMicrosoft Equation 3.0DS EquationEquation.39q,7(
ED=LFCompObjfObjInfoEquation Native 9_1271113090FZZOle
CompObjfObjInfoEquation Native 9
FMicrosoft Equation 3.0DS EquationEquation.39q,8(
ED=PF
FMicrosoft Equation 3.0DS EquationEquation.39q_1271113111FZZOle
CompObjfObjInfoEquation Native 9_1271113140FZZOle
CompObjfX*6(
PF=LF
FMicrosoft Equation 3.0DS EquationEquation.39qx) 
ME=DF
FMicrosoft Equation 3.0DS EqObjInfoEquation Native 9_1271113159FZZOle
CompObjfObjInfoEquation Native 9_1271113181FZZuationEquation.39q,!
ME=EP
FMicrosoft Equation 3.0DS EquationEquation.39qP,d#
EP=DFOle
CompObjfObjInfoEquation Native 9_1271113201FZZOle
CompObjfObjInfo"%()*,./01245678:
FMicrosoft Equation 3.0DS EquationEquation.39q, 
MED
FMicrosoft Equation 3.0DS EquationEquation.39qEquation Native 1_1271113223FZZOle
CompObjfObjInfoEquation Native 1_1271113244FZZOle
X*"
LFD
FMicrosoft Equation 3.0DS EquationEquation.39q,$
DM=DLCompObjfObjInfo
Equation Native 9_1270622742FZZOle
CompObjfObjInfoEquation Native
FMicrosoft Equation 3.0DS EquationEquation.39qo5
a1
,a2
,& ,a2004
FMicrosoft Equation 3.0DS Eq_1270622608FZZOle
CompObjfObjInfouationEquation.39qj<2
a1
e"a2
e""e"a2004
FMicrosoft Equation 3.0DS EquationEquation.39qEquation Native _1270622801FZZOle
CompObjfObjInfo Equation Native !_12706260061FZZOle
#c!6
a1
+a2
+"+a2004
d"1
FMicrosoft Equation 3.0DS EquationEquation.39q'<2%
a12
+CompObj$fObjInfo&Equation Native '1Table3a22
+5a32
+7a42
+"+4007a20042
d"1Oh+'0$8 T`
PADVANCED PROBLEM SOLVING ACTIVITIES: HOW DO STUDENTS NAVIGATE TERRA INCOGNITAOleksiy[^Rݹ+gT3G¬^Ψ1,GGnx};?ι,25\͠\g/טC!wy&7)x5p
'/E.o0/_/G7/^Tp
R[s2=e
/S%#myÿ#Y
?FC܁^1ArGΝ5^b~(cu^T`t,~e1:wG7/=^s6qw{K$/)8m7;xQ}R)y9C慟#7#^Tpvs4 sS6=} i?qȥvx.&y\(h
ŏr_j$/?}^T`</yh/#K:=~]tkPxz9gΉ>22/*_U,yQ}R/9"}tǆy7{glޓ+'yx*ٟE 6,/j? >1vODd
b
G
c$AF??3"`?;2$ӰN1x5^ui`!m$ӰN1x5^.xH;xcdd``gd``baV d,FYzP1n:x,56~) @ '00
ㆪaM,,He`0 @201W&0OCIA =i,@պ@@ڈ+уJt̻(ʥ13X?MaDd
b
H
c$AG??3"`?<2@>8}gi`!>8}g:hxcdd`` @c112BYL%bpu2?Wo;x6JtdXi`!Wo;x6JtdX:`0xcdd`` @c112BYL%bpu??3"`?@2jݧx6'#YF،i`!>ݧx6'#Yxcdd``> @c112BYL%bpuH
e*+Qr
HE7EI9~\'a"#5يd/\`?2M%F&&\1@]
U`lJDd
b
L
c$AK??3"`?A2?w*bs%i`!w*bs%:Rxcdd`` @c112BYL%bpu?O,@r0]Y!Hxcdd``> @c112BYL%bpuH4A#(Ff +9D&W&00ZQ Pdl:.v0o(T021)W2LԡRYz]`2NjCDd
b
B
c$AA??3"`?F2TC~{"}igi`!aTC~{"}TH/xcdd``bd``baV d,FYzP1n:,56~) @ '30
ㆪaM,,He`0 @201W&00j\Qc@`^*ZHq%0ܛXnZZk +ss%8~6a>#dO[7.o>fK#F&&\+@]
? 1,I_Dd
b
P
c$AO??3"`?H2?\S<04A3Yv˝i`!\S<04A3Yv:xcdd`` @c112BYL%bpu1@`2`ĩ9Dd
b
T
c$AS??3"`?M2@낍z!z/Ĝi`!낍z!z/Ĝ:`0xcdd`` @c112BYL%bpu @c112BYL%bpui`!Sb?'q>:xcdd`` @c112BYL%bpuґA*
gA)OfRńS`OSO^S{kwhLh^xa;kr^zDVb%9Nz;1z(ة({w/?)տ7Dd
b
Y
c$AX??3"`?S2@9_Kkrи$i`!9_Kkrи$:hxcdd`` @c112BYL%bpuY/Hd*+1qsI0D]]R{hGpenR~8
B8`d8&
3p Dc.p0mE
3LLJ% {: ̿020e^hP Dd
Xb
^
c$A]??3"`?Z2DjZiP:; i`!jZiP:;<xcdd`` @c112BYL%bpufK#F&&\1@]
@Y]`G!Dd
b
b
c$Aa??3"`?^2k?U(jܣGi`!??U(jܣL
xcdd`` @c112BYL%bpu!Dd
b
d
c$Ac??3"`?`2k*1>דjG6i`!?*1>דj@8
xcdd`` @c112BYL%bpuk +ssp0D@[W.o>fK#F&&\ s: @fbb#3X!J.!Dd
b
e
c$Ad??3"`?a2kk +sscER
tp? S26.C``ÈI)$5a
\w.ٟ`~IDd
Xb
f
c$Ae??3"`?b2E{tU`r!xi`!{tU`r<xcdd`` @c112BYL%bpun.\P{naEv=#RpeqIj.Dw.NB[3X?6!Dd
$b
h
c$Ag??3"`?d2k>`I{ũGmi`!?>`I{ũHD
xcdd`` @c112BYL%bpu
1(;]C8P.P5F\9A"F&P̌`ohFFldd 60nhK1 t{A727)?1WŇ@y>w;Ȉp{ް1B
Qp()Jd
'ڿl #spAcSN%LLJ%'@2u3tA4g!t?2elx~Dd
hb
2
c$A1??3"`?g25]SEMWi`!5]SEM
@jxcdd``ncd``baV d,FYzP1n:&B@?b x
ㆪaM,,He`H @201d++&1bQv S4mP.P56j`jX 1O?y12@]`s0sDL8"Ǚ"0'X927)?0@y>n
`v_Ƃʿ2!n7&T!
%E\Z+.hpBS;+KRsepgkǰ#Dd
b
3
c$A2??3"`?h2>`N%6l7"i`!>`N%6l7" `P&0xcdd`` @c112BYL%bpu?@ABCDEFGHIJKLV@V
Normal$dx7$8$a$CJ_HaJmH sH tHDA@DDefault Paragraph FontRiRTable Normal4
l4a(k(No List^O^CONFBullets)$
&F
hq^`qa$tH ^O^
PME Normal$d@x7$8$a$CJ_HaJmH sH tH\O\
PME Heading 1$$x@&a$5;CJ KH\aJ POP
PME Heading 2$$x@&a$ 5;\HOH
PME Heading 3$$@&a$5\DOD PME Quote!d^!CJaJNORNPME Author/Institution$a$NOaNPME Normal =0:CJ_HaJmH sH tHBbqBPME Heading 3 =0:5\<b<PME Quote =0:CJaJVOVPME References!d^!`CJaJ:O:aPME Abstract6]$T
l>(>>#+,45768$T
l>(>B
H_rQ !
Gm2'Q''1+5+6+9+:+>+?+A+B+C+D+E+G+t++.22::s::;<t<Q=I>@DDDHJJKKMM@PaPfTTVWY Z\%]^^^^cyeeiim3ooppstt:uuuKvvwCxx6yy2zzR{{}s}4~$7rtŃ/*
v_`abcdeύЍэҍӍԍՍ֍؍ٍڍۍ܍ݍލߍ)*+;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\ڎێ,QRSTUVWXYZ[\]^_`abcdЏя;<=MNOPQRSTUVWXYZ[\]^_`abcdefghijklmn'(>?cdefghijklmnopqrstuvёґӑԑՑ֑בّؑڑۑܑݑޑߑ
!"#$%&'()*+,./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRuv'(9:tuړۓPQ^_ݔޔ/0TUxy
*+<=wxݖޖSTab23WXwxŘƘ֘
>?XYhixyęřޙߙ ()89:;<=>qr#$^_ԛ՛:;uv&'QRǝȝ=>hiޞߞ34MNghϟП78QRklɠʠ٠ڠ
)*9:IJYZijyz̡#IJZtբ֢.G000000000000000000(000000000000000000(000(00(00(000000000(000(00(00(00(00(00(00(00(00(00000(000000(000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000 0 0 00 0 0 0 000 0 0 0 0 00Dw_!
GmE+G+t++.22::;<t<Q=I>@DDDHJJKKMM@PaPfTTV%]^^^^yeeiim3ooppstt:uuuvwCxx6yy2zzR{{}s}4~$7rŃ/
v_`abcdeI00K00K00K00O?K00I0 0I00I0
0!K0
0!K0 0K00"
K00!I00 I00K0S I00I00K00K00.,cK00K00K00j00+K00K00)K00)K00K00&K00&K00.K00.K00&K00+K00&K00&K0"0*K00&K00&K00&K00&K0'0,(/K0'0+K0'0)K0'0K0'0@0I0'0,K00&K00K00I00K00K00K00K00K0&0 K00K00K00K00K00xI00I00I00I00I00K00K00K00K0
0K0
0K0
0@0K00K00K00K0 0 K00K00K00K0M0
K00K00K00K0"0K0"0#K0"0K0"0@0I00I0Z000000076>ABWCCGLR
STPUVVWyY^def3g
h!ii
j{jkk?ll9m%oxxF5ɏ0VYZ[\^_`abcefghijklmnoqrstuvwxyz{}~53C\rԕEۖMjq6R_=ǥW]dpX
D
X
Z
d
x
z
0DFH\^}3'K'N' :4:6:<:P:R:Z:n:p::::::::::;;;';;;=;C;W;Y;t;;;;;;;;;;;;6JJJLJJJJJJJJ
KK:KNKPK9LMLOLLLLLLLMMM"M6M8M?@d>>(
\ d.9'
3 s"*?`
c$X99?d.9'z2
c$f"`"2!
2
<R)R)I *!
Z2
S*lZ2
SQ&'Z2
STpZ2
S3Z2
S!
#/Z2
S#$vZB
SD
1*2ZB
B
SD
"\ZB
SDTZB
SDTabZB
B
SD5ZB
SDVZW\ZB
SD@BZB
SDp*`ZB
B
SD
ZB
SD"pWZB
SD TZB
SDp$q$ZB
SD/%"&!ZB
B
SD"#e ZB
SD+"o#ZB
SD')ZB
SDt'(ZB
B
SD/%Z"&ZB
SD&t'8ZB
SDN#t$ZB
SD"Q"UZB
!B
SD !ZB
"
SD"Z#
#
<R)R)#
#.&
ZB
$B
SD"1#ZB
%
SD #ZB
&
SDT U#ZB
'
SD!#ZB
(
SDX( Y(#ZB
)
SD%!%#ZB
*
SD" "#Z
+
3z
Z
,
32@I #
Z

3ap
ZB
.
SDRZB
/
SDzRP$ZB
0
SDaZB
1
SDp@ZB
2
SD2!ZB
3
SDI !!
4
<R)R)4K.
H
5
# 5
H
6
#6
H
7
#
7
H
8
#8
NB
9
SDNB
:
SDNB
;
SD NB
<
SDNB
=
SDx
>
<R)R)>
B
S ?2'L'1+2+3+6+7+:+;+<+?+>vt9!>t5pt7l%t6 t<py yt=ylyt:rt9rt;rt8 t`9($:(;(,<(=(>(,?(@(lA(lB(PC(PD(TE(pF(<
G(WH(DI(lJ(5K(8L(TM( N(UO(P(
Q(R(S(TT(U(tV(pW(,tX(rY(,lZ(dk[(\(<](4^(_(F`(Ja(b(lc("d(te(f(g(Yh(i(j(mk(l_l(lm($n(o(@p(<q(r(s(dt('u((v( w(,1x(ty(4z({((d}(~(D(((>((O((l(>((_($W(((DF((_((o(d(((<((,((,rr#X'....34666ttvvvvwww1x1x9xCxCxyy%yyy6{6{@{}%%ׁׁaarbbjxӇӇڇЈЈوUaagMMU͌
!"#$%&'(*)+,./0123456789:;<=?>@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_")22\' ....34666ttvvvvwww7x;x;xIxIx#y'y'yyy?{J{J{}+//߁iivhll؇ވވ[fooS]]Ɍ
!"#$%&'(*)+,./0123456789:;<=?>@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_CY*urn:schemasmicrosoftcom:office:smarttagsmetricconverter9_*urn:schemasmicrosoftcom:office:smarttagsplace8`*urn:schemasmicrosoftcom:office:smarttagsCity9H*urn:schemasmicrosoftcom:office:smarttagsStateB^*urn:schemasmicrosoftcom:office:smarttagscountryregion=\*urn:schemasmicrosoftcom:office:smarttags PlaceType=]*urn:schemasmicrosoftcom:office:smarttags PlaceName?1. A ProductID`_^]\`_Y^\_]__`_^^_^_`_`H_`_H`__`H_`]_\``_\]_`_`_H`_H`_H`_HH___`H`_H`_`H`_`H_H_`_H`_^`]_\`_H\]_^_fgqLM8@:;IQ !!# #%%A&J&++3344567788^8m8o8s8 :7:8:;:Q:S:Z:q:s:{::~:::::::::::::;;;&;<;>;C;Z;[;_;`;c;t;;;;;;;;;;;;MBWBD(D/P6P U(UWWXX.X3X%]<]=]?]@]A]O]f]g]i]j]k]}]]]]]]]]]]]]]]]]]^^%^<^J^P^i^j^m^n^v^z^^^^^^^^^^^^__``+dBdCdFd\d^dgd~ddddddddddddddd ee`eweye~eiiUj\jjjoouuuuuuuuv$v7w?wxwwwwxx)y3yyyyyyyzzzzzR{Y{{{Zau{}
}}}s}y}}}4~:~ mst~Ńσ˄Մۄ?E*0{UWƊЊhopz(!֢
!
%
y}GNk!x!!d"Q'Z'G+P+$8%8:r::;2KhKLLMMMMNNOOOO[[%]^^^^^____
a
aEaIabbccccccdxekkapepppqqtu:upuv%vKvhvivvvv1wwwxwwwww0xxxyyHzhzzzz5{6{Q{C}"}=}s}}4~X~Y~y~^ $W$7jkÁցa'/QR*<=f^(()u~QR+,xTU!yz'0>G3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333Qrt++: :7:<:S:Z:q:s:::::::;;';>;C;Z;t;;;;;;;;;;JJ2KiKKK@PaPYZ\%]<]O]f]}]]]]]]]]^%^'^>^P^g^z^^^^^^^^+dBdGd^dgd~dddddddddde
e`eweeiiopttuuyy2zHz~4~яБRvUyy4ʠ8:JJZԢG :7:<:S:Z:q:::::::;;';>;C;Z;t;;;;;;;;%]<]O]f]}]]]]]]]]^%^'^>^P^g^z^^^^+dBdGd^dgd~dddddddddde`ewe,*Up!^`OJQJo(hH^`OJQJ^Jo(hHo ^ `OJQJo(hH^`OJQJo(hH^`OJQJ^Jo(hHoQQ^Q`OJQJo(hH!!^!`OJQJo(hH^`OJQJ^Jo(hHo^`OJQJo(hH,*U\ a XN#ZQx
`@&>_Q^A(8ThH!sK! """&'K)q*#z,1.6]/0h0i051E1t1_+2H2_2FO3I4)59e6>8'G8nd9W6;A<R<Ls<>?!?u?*B14BCzBC/CbC`Dx#EOEG`GhG*JXJd NOSNIRaR1S]T%VKBVXXUY]YzYMjZn\pJ]0
`kfb/dezkg h*h^i2lGYl.9nCpRqhqs7sDtv;vz,wux]xZoxO}4<42'1=VA&
&:gXv&!uk*QQuTb$ 0GXe80
d9s_'ou&+0avUzLTY$'{NWX.<ojCcg