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Linear Algebra and its Applications

Linear Algebra and its Applications432(2010)2089–2099Contents lists available at ScienceDirect Linear Algebra and its Applications j o u r n a l h o m e p a g e:w w w.e l s e v i e r.c o m/l o c a t e/l aaIntegrating learning theories and application-based modules in teaching linear algebraୋWilliam Martin a,∗,Sergio Loch b,Laurel Cooley c,Scott Dexter d,Draga Vidakovic ea Department of Mathematics and School of Education,210F Family Life Center,NDSU Department#2625,P.O.Box6050,Fargo ND 58105-6050,United Statesb Department of Mathematics,Grand View University,1200Grandview Avenue,Des Moines,IA50316,United Statesc Department of Mathematics,CUNY Graduate Center and Brooklyn College,2900Bedford Avenue,Brooklyn,New York11210, United Statesd Department of Computer and Information Science,CUNY Brooklyn College,2900Bedford Avenue Brooklyn,NY11210,United Statese Department of Mathematics and Statistics,Georgia State University,University Plaza,Atlanta,GA30303,United StatesA R T I C L E I N F O AB S T R AC TArticle history:Received2October2008Accepted29August2009Available online30September2009 Submitted by L.Verde-StarAMS classiﬁcation:Primary:97H60Secondary:97C30Keywords:Linear algebraLearning theoryCurriculumPedagogyConstructivist theoriesAPOS–Action–Process–Object–Schema Theoretical frameworkEncapsulated process The research team of The Linear Algebra Project developed and implemented a curriculum and a pedagogy for parallel courses in (a)linear algebra and(b)learning theory as applied to the study of mathematics with an emphasis on linear algebra.The purpose of the ongoing research,partially funded by the National Science Foundation,is to investigate how the parallel study of learning theories and advanced mathematics inﬂuences the development of thinking of individuals in both domains.The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted,in some students,a rich understanding of both domains and that had a mutually reinforcing effect.Furthermore,there is evidence that the deeper insights will contribute to more effective instruction by those who become high school math teachers and,consequently,better learning by their students.The courses developed were appropriate for mathematics majors,pre-service secondary mathematics teachers, and practicing mathematics teachers.The learning seminar focused most heavily on constructivist theories,although it also examinedThe work reported in this paper was partially supported by funding from the National Science Foundation(DUE CCLI 0442574).∗Corresponding author.Address:NDSU School of Education,NDSU Department of Mathematics,210F Family Life Center, NDSU Department#2625,P.O.Box6050,Fargo ND58105-6050,United States.Tel.:+17012317104;fax:+17012317416.E-mail addresses:william.martin@(W.Martin),sloch@(S.Loch),LCooley@ (L.Cooley),SDexter@(S.Dexter),dvidakovic@(D.Vidakovic).0024-3795/$-see front matter©2009Elsevier Inc.All rights reserved.doi:10.1016/a.2009.08.0302090W.Martin et al./Linear Algebra and its Applications432(2010)2089–2099Thematicized schema Triad–intraInterTransGenetic decomposition Vector additionMatrixMatrix multiplication Matrix representation BasisColumn spaceRow spaceNull space Eigenspace Transformation socio-cultural and historical perspectives.A particular theory, Action–Process–Object–Schema(APOS)[10],was emphasized and examined through the lens of studying linear algebra.APOS has been used in a variety of studies focusing on student understanding of undergraduate mathematics.The linear algebra courses include the standard set of undergraduate topics.This paper reports the re-sults of the learning theory seminar and its effects on students who were simultaneously enrolled in linear algebra and students who had previously completed linear algebra and outlines how prior research has inﬂuenced the future direction of the project.©2009Elsevier Inc.All rights reserved.1.Research rationaleThe research team of the Linear Algebra Project(LAP)developed and implemented a curriculum and a pedagogy for parallel courses in linear algebra and learning theory as applied to the study of math-ematics with an emphasis on linear algebra.The purpose of the research,which was partially funded by the National Science Foundation(DUE CCLI0442574),was to investigate how the parallel study of learning theories and advanced mathematics inﬂuences the development of thinking of high school mathematics teachers,in both domains.The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted,in some teachers,a richer understanding of both domains that had a mutually reinforcing effect and affected their thinking about their identities and practices as teachers.It has been observed that linear algebra courses often are viewed by students as a collection of deﬁnitions and procedures to be learned by rote.Scanning the table of contents of many commonly used undergraduate textbooks will provide a common list of terms such as listed here(based on linear algebra texts by Strang[1]and Lang[2]).Vector space Kernel GaussianIndependence Image TriangularLinear combination Inverse Gram–SchmidtSpan Transpose EigenvectorBasis Orthogonal Singular valueSubspace Operator DecompositionProjection Diagonalization LU formMatrix Normal form NormDimension Eignvalue ConditionLinear transformation Similarity IsomorphismRank Diagonalize DeterminantThis is not something unique to linear algebra–a similar situation holds for many undergraduate mathematics courses.Certainly the authors of undergraduate texts do not share this student view of mathematics.In fact,the variety ways in which different authors organize their texts reﬂects the individual ways in which they have conceptualized introductory linear algebra courses.The wide vari-ability that can be seen in a perusal of the many linear algebra texts that are used is a reﬂection the many ways that mathematicians think about linear algebra and their beliefs about how students can come to make sense of the content.Instruction in a course is based on considerations of content,pedagogy, resources(texts and other materials),and beliefs about teaching and learning of mathematics.The interplay of these ideas shaped our research project.We deliberately mention two authors with clearly differing perspectives on an undergraduate linear algebra course:Strang’s organization of the material takes an applied or application perspective,while Lang views the material from more of a“pure mathematics”perspective.A review of the wide variety of textbooks to classify and categorize the different views of the subject would reveal a broad variety of perspectives on the teaching of the subject.We have taken a view that seeks to go beyond the mathe-matical content to integrate current theoretical perspectives on the teaching and learning of undergrad-uate mathematics.Our project used integration of mathematical content,applications,and learningW.Martin et al./Linear Algebra and its Applications432(2010)2089–20992091 theories to provide enhanced learning experiences using rich content,student meta cognition,and their own experience and intuition.The project also used co-teaching and collaboration among faculty with expertise in a variety of areas including mathematics,computer science and mathematics education.If one moves beyond the organization of the content of textbooks weﬁnd that at their heart they do cover a common core of the key ideas of linear algebra–all including fundamental concepts such as vector space and linear transformation.These observations lead to our key question“How is one to think about this task of organizing instruction to optimize learning?”In our work we focus on the conception of linear algebra that is developed by the student and its relationship with what we reveal about our own understanding of the subject.It seems that even in cases where researchers consciously study the teaching and learning of linear algebra(or other mathematics topics)the questions are“What does it mean to understand linear algebra?”and“How do I organize instruction so that students develop that conception as fully as possible?”In broadest terms, our work involves(a)simultaneous study of linear algebra and learning theories,(b)having students connect learning theories to their study of linear algebra,and(c)the use of parallel mathematics and education courses and integrated workshops.As students simultaneously study mathematics and learning theory related to the study of mathe-matics,we expect that reﬂection or meta cognition on their own learning will enable them to construct deeper and more meaningful understanding in both domains.We chose linear algebra for several reasons:It has not been the focus of as much instructional research as calculus,it involves abstraction and proof,and it is taken by many students in different programs for a variety of reasons.It seems to us to involve important mathematical content along with rich applications,with abstraction that builds on experience and intuition.In our pilot study we taught parallel courses:The regular upper division undergraduate linear algebra course and a seminar in learning theories in mathematics education.Early in the project we also organized an intensive three-day workshop for teachers and prospective teachers that included topics in linear algebra and examination of learning theory.In each case(two sets of parallel courses and the workshop)we had students reﬂect on their learning of linear algebra content and asked them to use their own learning experiences to reﬂect on the ideas about teaching and learning of mathematics.Students read articles–in the case of the workshop,this reading was in advance of the long weekend session–drawn from mathematics education sources including[3–10].APOS(Action,Process,Object,Schema)is a theoretical framework that has been used by many researchers who study the learning of undergraduate and graduate mathematics[10,11].We include a sketch of the structure of this framework and refer the reader to the literature for more detailed descriptions.More detailed and speciﬁc illustrations of its use are widely available[12].The APOS Theoretical Framework involves four levels of understanding that can be described for a wide variety of mathematical concepts such as function,vector space,linear transformation:Action,Process,Object (either an encapsulated process or a thematicized schema),Schema(Intra,inter,trans–triad stages of schema formation).Genetic decomposition is the analysis of a particular concept in which developing understanding is described as a dynamic process of mental constructions that continually develop, abstract,and enrich the structural organization of an individual’s knowledge.We believe that students’simultaneous study of linear algebra along with theoretical examination of teaching and learning–particularly on what it means to develop conceptual understanding in a domain –will promote learning and understanding in both domains.Fundamentally,this reﬂects our view that conceptual understanding in any domain involves rich mental connections that link important ideas or facts,increasing the individual’s ability to relate new situations and problems to that existing cognitive framework.This view of conceptual understanding of mathematics has been described by various prominent math education researchers such as Hiebert and Carpenter[6]and Hiebert and Lefevre[7].2.Action–Process–Object–Schema theory(APOS)APOS theory is a theoretical perspective of learning based on an interpretation of Piaget’s construc-tivism and poses descriptions of mental constructions that may occur in understanding a mathematical concept.These constructions are called Actions,Processes,Objects,and Schema.2092W.Martin et al./Linear Algebra and its Applications432(2010)2089–2099 An action is a transformation of a mathematical object according to an explicit algorithm seen as externally driven.It may be a manipulation of objects or acting upon a memorized fact.When one reﬂects upon an action,constructing an internal operation for a transformation,the action begins to be interiorized.A process is this internal transformation of an object.Each step may be described or reﬂected upon without actually performing it.Processes may be transformed through reversal or coordination with other processes.There are two ways in which an individual may construct an object.A person may reﬂect on actions applied to a particular process and become aware of the process as a totality.One realizes that transformations(whether actions or processes)can act on the process,and is able to actually construct such transformations.At this point,the individual has reconstructed a process as a cognitive object. In this case we say that the process has been encapsulated into an object.One may also construct a cognitive object by reﬂecting on a schema,becoming aware of it as a totality.Thus,he or she is able to perform actions on it and we say the individual has thematized the schema into an object.With an object conception one is able to de-encapsulate that object back into the process from which it came, or,in the case of a thematized schema,unpack it into its various components.Piaget and Garcia[13] indicate that thematization has occurred when there is a change from usage or implicit application to consequent use and conceptualization.A schema is a collection of actions,processes,objects,and other previously constructed schemata which are coordinated and synthesized to form mathematical structures utilized in problem situations. Objects may be transformed by higher-level actions,leading to new processes,objects,and schemata. Hence,reconstruction continues in evolving schemata.To illustrate different conceptions of the APOS theory,imagine the following’teaching’scenario.We give students multi-part activities in a technology supported environment.In particular,we assume students are using Maple in the computer lab.The multi-part activities,focusing on vectors and operations,in Maple begin with a given Maple code and drawing.In case of scalar multiplication of the vector,students are asked to substitute one parameter in the Maple code,execute the code and observe what has happened.They are asked to repeat this activity with a different value of the parameter.Then students are asked to predict what will happen in a more general case and to explain their reasoning.Similarly,students may explore addition and subtraction of vectors.In the next part of activity students might be asked to investigate about the commutative property of vector addition.Based on APOS theory,in theﬁrst part of the activity–in which students are asked to perform certain operation and make observations–our intention is to induce each student’s action conception of that concept.By asking students to imagine what will happen if they make a certain change–but do not physically perform that change–we are hoping to induce a somewhat higher level of students’thinking, the process level.In order to predict what will happen students would have to imagine performing the action based on the actions they performed before(reﬂective abstraction).Activities designed to explore on vector addition properties require students to encapsulate the process of addition of two vectors into an object on which some other action could be performed.For example,in order for a student to conclude that u+v=v+u,he/she must encapsulate a process of adding two vectors u+v into an object(resulting vector)which can further be compared[action]with another vector representing the addition of v+u.As with all theories of learning,APOS has a limitation that researchers may only observe externally what one produces and discusses.While schemata are viewed as dynamic,the task is to attempt to take a snap shot of understanding at a point in time using a genetic decomposition.A genetic decomposition is a description by the researchers of speciﬁc mental constructions one may make in understanding a mathematical concept.As with most theories(economics,physics)that have restrictions,it can still be very useful in describing what is observed.3.Initial researchIn our preliminary study we investigated three research questions:•Do participants make connections between linear algebra content and learning theories?•Do participants reﬂect upon their own learning in terms of studied learning theories?W.Martin et al./Linear Algebra and its Applications432(2010)2089–20992093•Do participants connect their study of linear algebra and learning theories to the mathematics content or pedagogy for their mathematics teaching?In addition to linear algebra course activities designed to engage students in explorations of concepts and discussions about learning theories and connections between the two domains,we had students construct concept maps and describe how they viewed the connections between the two subjects. We found that some participants saw signiﬁcant connections and were able to apply APOS theory appropriately to their learning of linear algebra.For example,here is a sketch outline of how one participant described the elements of the APOS framework late in the semester.The student showed a reasonable understanding of the theoretical framework and then was able to provide an example from linear algebra to illustrate the model.The student’s description of the elements of APOS:Action:“Students’approach is to apply‘external’rules toﬁnd solutions.The rules are said to be external because students do not have an internalized understanding of the concept or the procedure toﬁnd a solution.”Process:“At the process level,students are able to solve problems using an internalized understand-ing of the algorithm.They do not need to write out an equation or draw a graph of a function,for example.They can look at a problem and understand what is going on and what the solution might look like.”Object level as performing actions on a process:“At the object level,students have an integrated understanding of the processes used to solve problems relating to a particular concept.They un-derstand how a process can be transformed by different actions.They understand how different processes,with regard to a particular mathematical concept,are related.If a problem does not conform to their particular action-level understanding,they can modify the procedures necessary toﬁnd a solution.”Schema as a‘set’of knowledge that may be modiﬁed:“Schema–At the schema level,students possess a set of knowledge related to a particular concept.They are able to modify this set of knowledge as they gain more experience working with the concept and solving different kinds of problems.They see how the concept is related to other concepts and how processes within the concept relate to each other.”She used the ideas of determinant and basis to illustrate her understanding of the framework. (Another student also described how student recognition of the recursive relationship of computations of determinants of different orders corresponded to differing levels of understanding in the APOS framework.)Action conception of determinant:“A student at the action level can use an algorithm to calculate the determinant of a matrix.At this level(at least for me),the formula was complicated enough that I would always check that the determinant was correct byﬁnding the inverse and multiplying by the original matrix to check the solution.”Process conception of determinant:“The student knows different methods to use to calculate a determinant and can,in some cases,look at a matrix and determine its value without calculations.”Object conception:“At the object level,students see the determinant as a tool for understanding and describing matrices.They understand the implications of the value of the determinant of a matrix as a way to describe a matrix.They can use the determinant of a matrix(equal to or not equal to zero)to describe properties of the elements of a matrix.”Triad development of a schema(intra,inter,trans):“A singular concept–basis.There is a basis for a space.The student can describe a basis without calculation.The student canﬁnd different types of bases(column space,row space,null space,eigenspace)and use these values to describe matrices.”The descriptions of components of APOS along with examples illustrate that this student was able to make valid connections between the theoretical framework and the content of linear algebra.While the2094W.Martin et al./Linear Algebra and its Applications432(2010)2089–2099descriptions may not match those that would be given by scholars using APOS as a research framework, the student does demonstrate a recognition of and ability to provide examples of how understanding of linear algebra can be organized conceptually as more that a collection of facts.As would be expected,not all participants showed gains in either domain.We viewed the results of this study as a proof of concept,since there were some participants who clearly gained from the experience.We also recognized that there were problems associated with the implementation of our plan.To summarize ourﬁndings in relation to the research questions:•Do participants make connections between linear algebra content and learning theories?Yes,to widely varying degrees and levels of sophistication.•Do participants reﬂect upon their own learning in terms of studied learning theories?Yes,to the extent possible from their conception of the learning theories and understanding of linear algebra.•Do participants connect their study of linear algebra and learning theories to the mathematics content or pedagogy for their mathematics teaching?Participants describe how their experiences will shape their own teaching,but we did not visit their classes.Of the11students at one site who took the parallel courses,we identiﬁed three in our case studies (a detailed report of that study is presently under review)who demonstrated a signiﬁcant ability to connect learning theories with their own learning of linear algebra.At another site,three teachers pursuing math education graduate studies were able to varying degrees to make these connections –two demonstrated strong ability to relate content to APOS and described important ways that the experience had affected their own thoughts about teaching mathematics.Participants in the workshop produced richer concept maps of linear algebra topics by the end of the weekend.Still,there were participants who showed little ability to connect material from linear algebra and APOS.A common misunderstanding of the APOS framework was that increasing levels cor-responded to increasing difﬁculty or complexity.For example,a student might suggest that computing the determinant of a2×2matrix was at the action level,while computation of a determinant in the 4×4case was at the object level because of the increased complexity of the computations.(Contrast this with the previously mentioned student who observed that the object conception was necessary to recognize that higher dimension determinants are computed recursively from lower dimension determinants.)We faced more signiﬁcant problems than the extent to which students developed an understanding of the ideas that were presented.We found it very difﬁcult to get students–especially undergraduates –to agree to take an additional course while studying linear algebra.Most of the participants in our pilot projects were either mathematics teachers or prospective mathematics teachers.Other students simply do not have the time in their schedules to pursue an elective seminar not directly related to their own area of interest.This problem led us to a new project in which we plan to integrate the material on learning theory–perhaps implicitly for the students–in the linear algebra course.Our focus will be on working with faculty teaching the course to ensure that they understand the theory and are able to help ensure that course activities reﬂect these ideas about learning.4.Continuing researchOur current Linear Algebra in New Environments(LINE)project focuses on having faculty work collaboratively to develop a series of modules that use applications to help students develop conceptual understanding of key linear algebra concepts.The project has three organizing concepts:•Promote enhanced learning of linear algebra through integrated study of mathematical content, applications,and the learning process.•Increase faculty understanding and application of mathematical learning theories in teaching linear algebra.•Promote and support improved instruction through co-teaching and collaboration among faculty with expertise in a variety of areas,such as education and STEM disciplines.W.Martin et al./Linear Algebra and its Applications432(2010)2089–20992095 For example,computer and video graphics involve linear transformations.Students will complete a series of activities that use manipulation of graphical images to illustrate and help them move from action and process conceptions of linear transformations to object conceptions and the development of a linear transformation schema.Some of these ideas were inspired by material in Judith Cederberg’s geometry text[14]and some software developed by David Meel,both using matrix representations of geometric linear transformations.The modules will have these characteristics:•Embed learning theory in linear algebra course for both the instructor and the students.•Use applied modules to illustrate the organization of linear algebra concepts.•Applications draw on student intuitions to aid their mental constructions and organization of knowledge.•Consciously include meta-cognition in the course.To illustrate,we sketch the outline of a possible series of activities in a module on geometric linear transformations.The faculty team–including individuals with expertise in mathematics,education, and computer science–will develop a series of modules to engage students in activities that include reﬂection and meta cognition about their learning of linear algebra.(The Appendix contains a more detailed description of a module that includes these activities.)Task1:Use Photoshop or GIMP to manipulate images(rotate,scale,ﬂip,shear tools).Describe and reﬂect on processes.This activity uses an ACTION conception of transformation.Task2:Devise rules to map one vector to another.Describe and reﬂect on process.This activity involves both ACTION and PROCESS conceptions.Task3:Use a matrix representation to map vectors.This requires both PROCESS and OBJECT conceptions.Task4:Compare transform of sum with sum of transforms for matrices in Task3as compared to other non-linear functions.This involves ACTION,PROCESS,and OBJECT conceptions.Task5:Compare pre-image and transformed image of rectangles in the plane–identify software tool that was used(from Task1)and how it might be represented in matrix form.This requires OBJECT and SCHEMA conceptions.Education,mathematics and computer science faculty participating in this project will work prior to the semester to gain familiarity with the APOS framework and to identify and sketch potential modules for the linear algebra course.During the semester,collaborative teams of faculty continue to develop and reﬁne modules that reﬂect important concepts,interesting applications,and learning theory:Modules will present activities that help students develop important concepts rather than simply presenting important concepts for students to absorb.The researchers will study the impact of project activities on student learning:We expect that students will be able to describe their knowledge of linear algebra in a more conceptual(structured) way during and after the course.We also will study the impact of the project on faculty thinking about teaching and learning:As a result of this work,we expect that faculty will be able to describe both the important concepts of linear algebra and how those concepts are mentally developed and organized by students.Finally,we will study the impact on instructional practice:Participating faculty should continue to use instructional practices that focus both on important content and how students develop their understanding of that content.5.SummaryOur preliminary study demonstrated that prospective and practicing mathematics teachers were able to make connections between their concurrent study of linear algebra and of learning theories relating to mathematics education,speciﬁcally the APOS theoretical framework.In cases where the participants developed understanding in both domains,it was apparent that this connected learning strengthened understanding in both areas.Unfortunately,we were unable to encourage undergraduate students to consider studying both linear algebra and learning theory in separate,parallel courses. Consequently,we developed a new strategy that embeds the learning theory in the linear algebra。

TIMSS、PISA、NAEP科学测试框架与测试题目的比较研究

TIMSS、PISA、NAEP科学测试框架与测试题目的比较研究姓名：梁润婵导师：李宏翰博士年级：2006级专业：课程与教学论方向：科学教育摘要进行教育评价，是教育活动中的一个重要方面。

但因其评价对象的复杂性，对教育评价的研究一直以来都是人们争论的热点。

考试作为评价的重要手段之一，是目前进行大规模教育评价的主要方式。

近年来，我国也开始在一些省市推行一些大规模学业评价项目。

但是如何确定考试的测量目标以及如何设计出高质量的试题来完成对测量目标的测量？尤其是对于实践性较强的科学学科，对学生科学能力的考查应包含哪些方面？如何用纸笔测验来对这些科学能力进行评价？都是当前亟待解决的问题。

本研究对国际上现行的三大教育评价项目TIMSS2007、PISA2006和NAEP2000&2005的八年级科学的测试框架和测试题目进行了分析和比较，旨在发现它们在进行科学评价时的理念以及实现这些理念所采用的方式，期望能对我国的大规模考试提供一些线索和帮助。

为达到本研究的目的，特确立了如下研究内容：（1）对测试框架的研究其中包括：评价目的、评价设计的哲学基础、评价中各内容领域所占的比重、评价中各认知能力所占的比重。

（2）对测试题目的研究其中包括：试卷结构、试卷难度和对科学探究的考查。

其中试卷结构又包括：试卷的长度（或题量）、题型以及不同题型试题的比例。

其中试卷难度包括：情境与不同情境试题所占的比重、对多步推理能力的要求及不同要求所占的比例。

比较的过程分几个阶段：确定比较的问题；确定比较的标准；收集资料并加进行分析、解释；比较分析和结论。

其中在确立试卷难度的比较标准时，借鉴了David Nohara和Arnold A.Goldstein在2001年6月撰写的《NAEP、TIMSS-R和PISA的比较研究》。

经过以上研究过程，我们得到了如下结论：1．测试框架方面（1）在评价目的方面，TIMSS与NAEP比较接近。

其中PISA是在教育框架之外对教育成效的评价。

关于伟大科学家英语作文演讲稿高中

关于伟大科学家英语作文演讲稿高中全文共3篇示例，供读者参考篇1Great Minds of Science: Pioneering Discoveries and Lasting ImpactsHello everyone, today I'll be discussing some of the most brilliant and influential scientists throughout history. These innovative thinkers have shaped our understanding of the world and universe around us through ground-breaking theories, experiments, and discoveries. Their work has propelled humanity forward and expanded the frontiers of knowledge.Let's start with arguably the most famous scientist of all time, Sir Isaac Newton. This 17th century English mathematician, physicist, astronomer and natural philosopher is best known for his three laws of motion and his law of universal gravitation. Newton's revolutionary ideas like his theory that white light is made up of a spectrum of colors laid the foundation for the field of modern physical optics. His principle of natural philosophy, known as Newton's laws, were instrumental in advancing our comprehension of the physical forces that govern nature. He wasalso the inventor of calculus, although there is some debate over whether he developed the concepts independently from the German philosopher Gottfried Leibniz. Nonetheless, Newton's immense contributions can't be overstated - he is considered one of the most influential scientists in human history.Moving ahead a century later, we encounter the brilliant mind of Michael Faraday. As a self-taught English scientist, Faraday is responsible for several pivotal discoveries in the fields of electromagnetism and electrochemistry. He discovered the principles of electromagnetic induction and laws of electrolysis. Faraday also invented some of the first electric motors and developed the concepts that later became known as fields in theoretical physics. His experiments with electricity laid the groundwork for how we generate and utilize electrical power today. Despite having little formal education, Faraday's innate curiosity, diligence, and genius allowed him tomake monumental strides in the scientific world.No discussion about great scientists would be complete without mentioning the famous German-born theoretical physicist, Albert Einstein. He is most renowned for his theories of special and general relativity, which fundamentally changed our perception of space, time, energy, and gravity. Einstein proposedthe equation E=mc^2, which shows that tiny amounts of mass can be converted into enormous amounts of energy. His work also predicted the existence of gravitational waves, which were finally detected experimentally in 2015, over a century after he theorized them. Einstein received the Nobel Prize in 1921, and his theories and discoveries transformed physics and led to world-changing technologies like nuclear power, lasers, and fiber optics. He is considered one of the two pivotal intellects of the 20th century, along with Isaac Newton.While Einstein was blazing trails in physics, another scientific mastermind was making waves in the field of chemistry. Marie Curie, the first female professor at the University of Paris, conducted pioneering research on radioactivity. Together with her husband Pierre, the Polish-born physicist and chemist discovered the radioactive elements radium and polonium, advancing the study of atomic theory. Curie was also the first person to win two Nobel Prizes - one in Physics in 1903, which she shared with Pierre and Henri Becquerel, and one in Chemistry in 1911. Her work led to the development of X-rays and laid the foundations for the treatment of cancer using radiation. Despite facing immense prejudice as a woman in a male-dominated field, Marie Curie's brilliance and determinationallowed her to pursue scientific inquiries of profound importance.Fast forwarding to the mid-20th century, we arrive at the trailblazing computer scientist Alan Turing. This English mathematician and codebreaker is considered the father of theoretical computer science and artificial intelligence. During World War II, he made key contributions in cracking the Nazi's Enigma code with his pioneering machine at Britain's Bletchley Park facility. Turing's 1936 paper laid out the mathematical underpinnings of computability and introduced the concept of a theoretical computing machine called the "Turing machine." His ideas were hugely influential in the development of modern computers and his test for evaluating artificial intelligence is still used today. Turing was prosecuted for his homosexuality in the 1950s by the British government in a devastating injustice. However, his lasting legacy as a visionary scientist lives on.These are just a few examples of the innovative geniuses who have dramatically reshaped scientific understanding through their monumental discoveries and advancements. From Newton's laws of motion and universal gravitation, to Faraday's early electric motors, to Einstein's theories of relativity and nuclear physics, to Curie's work on radioactivity and radium, toTuring's foundations for computer science and AI - the contributions of these great thinkers have been immeasurable. They persevered through challenges, followed their curiosity, and pushed past the boundaries of conventional wisdom to forge new paths of knowledge.Science relies on this human drive to question, theorize, hypothesize, experiment, and explore the unknown. It is built on the intellectual courage to conceive radical ideas that may initially seem outlandish or implausible. The great scientistswe've discussed today all persisted despite facing criticism, setbacks and even oppression. Yet their pioneering spirits allowed them to revolutionize our comprehension of the physical world and universe.As students, we have the opportunity to follow in their footsteps as champions of scientific inquiry and critical thinking. We can be inspired by their examples to approach problems with creativity, logic and an inquisitive mindset. The more we understand the trailblazing work of those who came before, the better equipped we'll be to build upon their insights and continue expanding the frontiers of human knowledge.So let us celebrate these scientific pioneers and strive to emulate their intellectual bravery, diligence and passion forunveiling the mysteries of our cosmos. For it is this undying quest for knowledge and truth that enriches our existence and allows humanity to progress. As Sir Isaac Newton himself once declared, "No great discovery was ever made without a bold guess." The great minds of science were willing to make those bold conjectures and change the course of history. What bold guesses will you make to shape our future?篇2Great Minds that Shaped Our World: A Tribute to Pioneering ScientistsEsteemed guests, faculty, and my fellow students, today I stand before you humbled by the remarkable individuals whose extraordinary contributions have profoundly impacted our understanding of the universe and the world we inhabit.Throughout human history, certain minds have shone brilliantly, illuminating the path of progress and reshaping our collective knowledge. These great scientists, driven by an insatiable curiosity and an unwavering pursuit of truth, have challenged conventional wisdom, defied limitations, and ushered in paradigm shifts that have left an indelible mark on humanity.One such luminary whose legacy continues to inspire awe is Sir Isaac Newton. With his groundbreaking work on the laws of motion, universal gravitation, and the foundation of modern physics, Newton revolutionized our comprehension of the physical world. His seminal work, the Principia Mathematica, stands as a testament to his genius, a masterpiece that unveiled the intricate dance of celestial bodies and the fundamental forces that govern their movements.Yet, Newton's brilliance was not confined to the realms of physics alone. His contributions to mathematics, particularly in the development of calculus, opened new frontiers in the quantitative analysis of change and motion. His methodical approach and unwavering dedication to empirical evidence laid the groundwork for the scientific revolution, forever altering the course of human inquiry.Another titan whose name resonates through the ages is Charles Darwin. His theory of evolution by natural selection, a profound and controversial idea at the time, transformed our understanding of the origin and diversity of life on our planet. Through meticulous observation and analysis, Darwin proposed a mechanism that explained the gradual adaptation and transformation of species over vast expanses of time.His seminal work, "On the Origin of Species," challenged long-held beliefs and sparked heated debates within the scientific community and beyond. Yet, Darwin's unwavering pursuit of evidence and his willingness to confront the prevailing dogma of his era exemplified the essence of scientific inquiry – a relentless quest for truth, unencumbered by dogma or authority.As we stand on the shoulders of these giants, we cannot overlook the pioneering spirit of Marie Curie, a trailblazer whose accomplishments transcended gender barriers and shattered societal norms. Curie's groundbreaking work on radioactivity, which led to the discovery of two new elements – radium and polonium – ushered in a new era of understanding the atomic world.Her unwavering dedication, coupled with her remarkable intellect, earned her a place in history as the first woman to win a Nobel Prize and the first person to be awarded the prestigious honor twice. Curie's life and work serve as a powerful testament to the transformative power of perseverance and the ability of science to transcend boundaries.In the realm of theoretical physics, Albert Einstein stands as a towering figure, a visionary whose genius reshaped our understanding of the fundamental nature of reality itself. Histheory of relativity, which revolutionized our conception of space, time, energy, and gravity, shattered long-held notions and ushered in a new era of scientific thought.Einstein's unwavering curiosity and his willingness to challenge the established paradigms of his time led him to profound insights that continue to captivate and inspire generations of scientists. His iconic equation, E=mc^2, has become a symbol of scientific elegance and a testament to the power of human intellect to unravel the deepest mysteries of the cosmos.As we pay tribute to these extraordinary minds, we must also acknowledge the countless others whose contributions, though lesser-known, have played a pivotal role in shaping our modern world. From the pioneers of vaccination and antibiotics to the architects of the digital revolution, these individuals have tackled seemingly insurmountable challenges, driven by an insatiable thirst for knowledge and a commitment to improving the human condition.Their stories serve as a powerful reminder that greatness often emerges from humble beginnings, fueled by an unwavering passion for discovery and a relentless pursuit of understanding. It is this very spirit that has propelled humanityforward, enabling us to conquer diseases, unlock the secrets of the universe, and push the boundaries of what was once thought impossible.As students of science, we stand on the shoulders of these giants, inheriting a profound legacy and a responsibility to carry the torch of inquiry forward. It is our duty to honor their achievements by continuing to ask questions, to challenge assumptions, and to fearlessly explore the unknown frontiers of knowledge.We must embrace the spirit of scientific inquiry, fostering an environment where curiosity is nurtured, where diverse perspectives are welcomed, and where failure is not a deterrent but a stepping stone towards progress. For it is through this relentless pursuit of understanding that we will unravel the mysteries that remain, unlock new realms of possibility, and forge a path towards a future where knowledge knows no bounds.In this great endeavor, we must remember that science is not merely a collection of facts or theories; it is a living, breathing entity that evolves and adapts as new evidence emerges. We must remain open to paradigm shifts, embracing the unknownwith humility and a willingness to challenge our own preconceptions.As we stand on the precipice of new discoveries, let us draw inspiration from the lives and legacies of the great scientists who have come before us. Let their unwavering curiosity, their relentless pursuit of truth, and their unwavering dedication to the betterment of humanity guide our steps forward.For it is through their examples that we learn that greatness is not measured by accolades or recognition, but by the enduring impact of one's contributions and the unwavering commitment to pushing the boundaries of human knowledge.Today, we celebrate these visionaries, these pioneers, these architects of our modern world. Their names may be etched in the annals of history, but their legacy lives on in the countless lives they have touched, the mysteries they have unraveled, and the boundless potential they have unleashed.As we look to the future, let us embrace the mantle of scientific inquiry with reverence and a sense of profound responsibility. For it is through our collective efforts, our unwavering dedication, and our steadfast commitment to the pursuit of knowledge that we will continue to shape the world,unlocking the secrets of the universe and paving the way for a brighter, more enlightened tomorrow.Thank you.篇3Great Minds of Science: Inspiring Curiosity and InnovationHello everyone, today I want to talk about some of the greatest scientific minds throughout history - individuals whose curiosity, perseverance, and intellect have fundamentally shaped our understanding of the world and universe we live in.Let's start with a name that is practically synonymous with genius - Albert Einstein. This German-born theoretical physicist is best known for his theory of relativity, which revolutionized our comprehension of space, time, energy, and gravity. His famous equation E=mc^2 unlocked new realms of physics and paved the way for groundbreaking fields like nuclear energy.But Einstein was more than just a brilliant mind. He was a man of great wisdom and foresight, once stating: "The important thing is not to stop questioning. Curiosity has its own reason for existing." This insatiable curiosity drove him to challenge conventional thinking and develop radical new ideas that stood the test of time.Next, let's look at the incredible Marie Curie, the first woman to win a Nobel Prize and the only person in history to win the prestigious award twice in two different scientific fields - physics and chemistry. Despite facing immense discrimination and opposition as a woman in the male-dominated sciences, Curie persevered. Her pioneering work on radioactivity, including the discovery of the elements radium and polonium, opened up new frontiers in our understanding of matter and energy.Curie's resilience and passion for science still inspire young men and women today to pursue their dreams, regardless of obstacles. As she once said: "Nothing in life is to be feared, it is only to be understood. Now is the time to understand more, so that we may fear less."Moving on, I'd like to highlight the brilliance of the English mathematician, physicist, computer scientist, and all-around genius Alan Turing. His seminal work on artificial intelligence, computing machinery, and codebreaking during World War II cemented his legacy as one of the most influential minds of the 20th century.The "Turing test" he proposed remains a vital concept in the field of artificial intelligence, pushing us to create machines that can not only calculate but truly think. His ideas foreshadowedthe radical technological advancements we've witnessed in computing and AI. As Turing himself stated: "We can only see a short distance ahead, but we can see plenty there that needs to be done."No discussion of scientific greats would be complete without mentioning the Renaissance genius Leonardo da Vinci. While widely celebrated for his iconic artworks like the Mona Lisa and The Last Supper, da Vinci's incredible mind also delved deep into the realms of science, mathematics, engineering, anatomy, and more.His curiosity knew no bounds, driving him to meticulously study everything from the flow of water to the mechanics of flight. Da Vinci's abilities to observe, question, and innovate continue to inspire scientists and creative thinkers centuries later. As he wisely noted: "Learning never exhausts the mind."Finally, let's look to the stars and the groundbreaking work of Galileo Galilei. This Italian astronomer's pioneering use of the telescope and advocacy of the heliocentric model, which placed the Sun at the center of our solar system, cemented his place as a true revolutionary in the field.Despite facing immense opposition from the church and its doctrine, Galileo stood firm in his convictions and scientificobservations. He embodied the spirit of questioning perceived norms, once stating: "I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use."These are just a few of the countless scientific luminaries who have expanded the frontiers of human knowledge through their dedication, curiosity, and brilliance. Their stories remind us that progress is born from questioning the status quo, taking bold leaps, and never losing that childlike sense of wonder about the universe.As a student, I find these great minds incredibly inspiring. They teach us that intelligence is not some fixed trait, but something that can grow through relentless effort,open-mindedness, and an insatiable thirst for learning. If we approach the world with curiosity and determination like Einstein, Curie, Turing, da Vinci, and Galileo, who knows whatmind-bending discoveries and innovations we might unlock?The path of science is not always easy. It requires grit, creativity, and a willingness to challenge conventional wisdom. But by following in the footsteps of these giants, embracing our innate sense of wonder, and daring to ask daring questions, we can continue pushing the boundaries of human understanding.As Carl Sagan eloquently stated: "Somewhere, something incredible is waiting to be known." Let us honor the great scientific minds by maintaining that spirit of awe, curiosity, and intellectual bravery. For it is those very qualities that will propel humanity towards the next groundbreaking revelations and life-changing innovations.Thank you.。

halliday

Halliday and His Systemic-Functional GrammarAbstract:Michael Alexander Kirkwood Halliday (often M.A.K. Halliday) is the founder of Systemic-Functional Grammar which is one of the two world's major linguistics.He developed the internationally influential model of language,the systemic functional grammar . His grammatical descriptions go by the name of systemic functional grammar (SFG). Halliday describes language as a semiotic system, "not in the sense of a system of signs, but a systemic resource for meaning". For Halliday, language is a "meaning potential"; by extension, he defines linguistics as the study of "how people exchange meanings by 'languaging'".Halliday describes himself as a generalist, meaning that he has tried "to look at language from every possible vantage point", and has described his work as "wander[ing] the highways and byways of language".However, he has claimed that "to the extent that I favoured any one angle, it was the social: language as the creature and creator of human society".In addition to English, the model has been applied to other languages, both Indo-European and non-Indo-European.KeyWords: Halliday and his life,linguistic theory and description ,influences,relations to other branches of grammarMichael Halliday was born in Yorkshire, England in 1925. He was trained in Chinese for war service with the British army. He got his BA in Chinese language and literature at London University in 1947. From 1947 to 1949 he studied under the supervision of Luo Changpei at Peking University. From 1949 to 1950 he studied at Lingnan University, South China, tutored by Wang Li. On returning to England, he worked for his PhD degree under the supervision of J. R. Firth. By 1955, he finished his doctoral dissertation“The Language of the Chinese Secret History of the Mongols”, an analysis of a work written in a northern Chinese dialect in the 14th century. In the ten years that followed, he taught linguistics at Cambridge University and EdinburghUniversity. From 1965 to 1970, he was professor of linguistics in London University. Later, he taught in various places in theworld, including the United States. In 1975, he moved to Australia and founded the Department of Linguistics in the University of Sydney. He holds three honorary doctorates, and some professorships and fellowships in many institutions over the world. In 1995 he became guest professor of Peking University in China. In 1976 he moved to Australia as Foundation Professor of Linguistics at the University of Sydney, where he remained until he retired. He has worked in various regions of language study, both theoretical and applied, and has been especially concerned with applying the understanding of the basic principles of language to the theory and practice of education. He received the status of Emeritus Professor of the University of Sydney and Macquarie University, Sydney, in 1987, and is currently Distinguished Visiting Professor in the Faculty of Education, University of Hong Kong.In fact,Halliday has a wide range of research interests: semantics and grammar ofmodern English; language development in early childhood; text linguistics and register variation; educational applications of linguistics; text generation in artificialintelligence;Chinese grammar and phonology. He has produced works on a variety of topics covering systemic grammar and functional grammar; language acquisition; language and society; language and semiotics; language teaching and translation; text analysis and stylistics.However,among them the most important is Systemic Grammar and Functional Grammar. Influnced by his tutor Firth who thinks a system is a set of mutually exclusive options that come into play at some point in a linguistic structure. Like Firthian phonology, it is primarily concerned with the semantically relevant choices in the language as a whole.Halliday defines system as“a set of options, a set of possibilities A, B orC, together with a condition of entry”.On a very general leve l, there is the chain system and the choice system.In the system network, people are making choices. Halliday believes that there are realization relationships between various levels. Meaning is coded by wording which is coded by sound(or writing).How does Halliday‟s systemic grammar different from other schools? First, it attached great importance to the sociological aspects of language. Second, it views language as a form of doing rather than a form of knowing.Thirdly, it gives a relatively high priority to description of the characteristics of particular languages and particular varieties of language. Fourthly, it explains a number of aspects of language in terms of clines (ie. ungrammatical-more unusual-less usual-grammatical). Fifthly, it seeks verification of its hypotheses by means of observation from texts and by means of statistical techniques. Lastly, it has as its central category the category of the system.Anyway,Halliday has made a great contribution to the development of linguistics.Although due to the deficiency of human knowledge about language in genera,he failed to deal with the problem of meaning.His theory of Systemic Grammar and Functional Grammar has indeed brought something new to the lingustic world.Linguistic theory and descriptionThe grammar of experience: the cover of An Introduction to Functional Grammar, 2nd ed. (1994), by M.A.K. Halliday, showing the types of process as they have evolved in English grammar.Halliday is notable for his grammatical theory and descriptions, outlined in his book An Introduction to Functional Grammar, first published in 1985. A revised edition was published in 1994, and then a third, in which he collaborated with Christian Matthiessen, in 2004. But Ha lliday‟s conception of grammar -or …lexicogrammar‟ (a term he coined to argue that lexis and grammar are part of the same phenomenon) – is based on a more general theory of language as a social semiotic resource, or a …meaning potential‟ (systemic functional linguistics). Halliday follows Hjelmslev and Firth in distinguishing theoretical from descriptive categories in linguistics. He argues that …theoretical categories, and their inter-relations, construe an abstract model of language,they are interlocking and mutally defining. The theoretical architecture derives from work on the description of natural discourse, and as such …no very clear line is drawn between …(theoretical) linguistics‟ and …applied linguistics‟.Thus, the theory …is continually evolving as it is brought to bear on solving problems of a research or practical nature‟.Halliday contrasts theoretical categories with descriptive categ ories, defined as …categories set up in the description of particular languages‟. His descriptive work has been focusedon English and Chinese.Halliday rejects explicitly the claims about language associated with the generative tradition. Language, he argues, "cannot be equated with 'the set of all grammatical sentences', whether that set is conceived of as finite or infinite". He rejects the use of formal logic in linguistic theories as "irrelevant to the understanding of language" and the use of such approaches as "disastrous for linguistics". On Chomsky specifically, he writes that "imaginary problems were created by the whole series of dichotomies that Chomsky introduced, or took over unproblematized: not only syntax/semantics but also grammar/lexis, language/thought, competence/performance. Once these dichotomies had been set up, the problem arose of locating and maintaining the boundaries between them."Studies of grammarHalliday's first major work on the subject of grammar was "Categories of the theory of grammar", published in the journal Word in 1961. In this paper, he argued for four "fundamental categories" for the theory of grammar: unit, structure, class, and system. These categories, he argued, are "of the highest order of abstraction", but he defended them as those necessary to "make possible a coherent account of what grammar is and of its place in language" In articulating the category unit, Halliday proposed the notion of a rank scale. The units of grammar formed a "hierarchy", a scale from "largest" to "smallest" which he proposed as: "sentence", "clause", "group/phrase", "word" and "morpheme". Halliday defined structure as "likeness between events in successivity" and as "an arrangement of elements ordered in places'. Halliday rejects a view of structure as "strings of classes, such as nominal group + verbalgroup + nominal group" among which there is just a kind of mechanical solidarity" describing it instead as "configurations of functions, where the solidarity is organic".[27] Grammar as systemicHalliday's early paper shows that the notion of "system" has been part of his theory from its origins. Halliday explains this preoccupation in the following way: "It seemed to me that explanations of linguistic phenomena needed to be sought in relationships among systems rather than among structures –in what I once called "deep paradigms" –since these were essentially where speakers made their choices".[28] Halliday's "systemic grammar" is a semiotic account of grammar, because of this orientation to choice. Every linguistic act involves choice, and choices are made on many scales. Systemic grammars draw on system networks as their primary representation tool as a consequence. For instance, a major clause must display some structure that is the formal realization of a choice from the system of "voice", i.e. it must be either "middle" or "effective", where "effective" leads to the further choice of "operative" (otherwise known as 'active') or "receptive" (otherwise known as "passive").Grammar as "functional"Halliday's grammar is not just systemic, but systemic functional. He argues that the explanation of how language works "needed to be grounded in a functional analysis, since language had evolved in the process of carrying out certain critical functions as human beings interacted with their 'eco-social' environment".Halliday's earlygrammatical descriptions of English, called "Notes on Transitivity and Theme in English – Parts 1–3"include reference to "four components in the grammar of English representing four functions that the language as a communication system is required to carry out: the experiential, the logical, the discoursal and the speech functional or interpersonal". The "discoursal" function was renamed the "textual function".In this discussion of functions of language,Halliday draws on the work of Bühler and Malinowski. Halliday's notion of language functions,or "metafunctions", became part of his general linguistic theory.Relation to other branches of grammarHalliday's theory sets out to explain how spoken and written texts construe meanings and how the resources of language are organised in open systems and functionally bound to meanings. It is a theory of language in use, creating systematic relations between choices and forms within the less abstract strata of grammar and phonology, on the one hand, and more abstact strata such as context of situation and context of culture on the other. It is a radically different theory of language from others which explore less abstract strata as autonomous systems, the most notable being Noam Chomsky's. Since the principal aim of systemic functional grammar is to represent the grammatical system as a resource for making meaning, it addresses different concerns. For example, it does not try to address Chomsky's thesis that there is a "finite rule system which generates all and only the grammatical sentences in a language".Halliday's theory encourages a more open approach to the definition of language as a resource; rather than focus on grammaticality as such, a systemic functional grammatical treatment focuses instead on the relative frequencies of choices made in uses of language and assumes that these relative frequencies reflect the probability that particular paths through the available resources will be chosen rather than others. Thus, SFG does not describe language as afinite rule system, but rather as a system, realised by instantiations, that is continuously expanded by the very instantiations that realise it and that is continuously reproduced and recreated with use.Another way to understand the difference in concerns between systemic functional grammar and most variants of generative grammar is through Chomsky's claim that "linguistics is a sub-branch of psychology". Halliday investigates linguistics more as a sub-branch of sociology. SFG therefore pays much more attention to pragmatics and discourse semantics than is traditionally the case in formalism.The orientation of systemic functional grammar has served to encourage several further grammatical accounts that deal with some perceived weaknesses of the theory and similarly orient to issues not seen to be addressed in more structural accounts. Examples include the model of Richard Hudson called word grammar.InfluencesHalliday describes his grammar as built on the work of Saussure, Louis Hjelmslev, Malinowski, J.R. Firth, and the Prague school linguists. In addition, he drew on the work of the American anthropological linguists Boas, Sapir and Whorf. His "main inspiration" was Firth, to whom he owes, among other things, the notion of language as system. Among American linguists,Benjamin Lee Whorf had "the most profoundeffect on my own thinking". Whorf "showed how it is that human beings do not all mean alike, and how their unconscious ways of meaning are among the most significant manifestations of their culture" [6]From his studies in China, he lists Luo Changpei and Wang Li as two scholars from whom he gained "new and exciting insights into language". He credits Luo for giving him a diachronic perspective and insights into a non-Indo-European language family. From Wang Li he learnt "many things, including research methods in dialectology, the semantic basis of grammar, and the history of linguistics in China".[6]References( 1).Halliday, M.A.K. and Matthiessen, C.M.I.M. 2004. An Introduction to Functional Grammar. Arnold. p37ff.(2)Halliday, M.A.K. Applied linguistics as an evolving theme[A].AILA Gold Medal Speaker,Book of Abstracts[M]. AILA, Singapore, 2002.(4) Halliday, M.A.K. &Hasan, nguage,Text and Context[M]. Geelong, Vic. Deakin University Press,1985:89.(5) Kramsch, C.Context and Culture in Language Teaching[M].Oxfor(6),(Halliday, 1994:xxvi):(7). Details of Halliday's work history from "M.A.K. Halliday" in Keith Brown and Vivien Law (eds).。

日常英语怎样说

日常英语怎样说日常指平时的,经常的，每日都要做的事情等。

那么你知道日常用英语怎么说吗?下面来学习一下吧。

日常英语说法1：daily日常英语说法2：usual日常英语说法3：everyday日常的相关短语：日常问候Everyday Greetings ; Daily Greetings ; Daily greetings to the students日常检查routine inspection ; Daily inspection ; current control ; current check日常概念everyday concept ; common conception ; daily concept ; common concept日常纳税 Daily tax payment日常交流Conversations among Family Members ; conmunication between each other ; Daily Communication 日常运作 day to day operations ; daily operation ; operational work日常事物 routine日常的英语例句：1. All too often they become enmeshed in deadening routines.他们时常陷入枯燥的日常事务之中。

2. Of course, the British will suffer such daily stresses patiently.当然，英国人将会耐心地承受这些日常压力。

3. Dietary experts can advise on the ins and outs of dieting.饮食专家可以对日常饮食的方方面面提出建议。

4. He continued to mow the lawn and do other routine chores.他继续剪草坪，并做些其他日常杂务。

项目指南发育谱系

项目指南发育谱系Developmental Spectrum.The developmental spectrum refers to the range of developmental abilities and behaviors observed in individuals. It encompasses the typical patterns of growth and development as well as variations that fall outside of these norms. Understanding the developmental spectrum is important for identifying and supporting individuals with developmental challenges.Early Development.Early development encompasses the period fromconception to early childhood. During this time,individuals experience rapid growth and change in all areas, including physical, cognitive, social, and emotional development. Key milestones during this period includemotor development, language acquisition, and the development of social and emotional bonds.Variations in Development.Variations in development can occur for a variety of reasons, including genetic factors, environmental influences, and preand perinatal complications. These variations can affect any aspect of development and can range from mild to severe. Some common examples of developmental variations include:Delayed development: Individuals may reach developmental milestones later than expected.Developmental disorders: These are more severe and persistent patterns of developmental delays or impairments that affect multiple areas of development. Examples include autism spectrum disorder, intellectual disability, and cerebral palsy.Learning disabilities: These are specific difficulties in acquiring and using academic skills, such as reading, writing, or mathematics.Behavioral problems: These may include aggression, hyperactivity, or difficulty with social interactions.Supporting Individuals with Developmental Challenges.Individuals with developmental challenges may require additional support and services to help them reach their full potential. This support can include:Early intervention: Services that provide support to young children with developmental delays or disabilities.Special education: Educational services designed to meet the unique needs of students with disabilities.Therapy: Occupational therapy, physical therapy, and speech therapy can help improve motor skills, language development, and other areas of functioning.Counseling: This can provide emotional support and guidance to individuals and families coping withdevelopmental challenges.Importance of Understanding the Developmental Spectrum.Understanding the developmental spectrum is essential for:Identifying and diagnosing developmental challenges. Providing appropriate support and services.Promoting inclusion and acceptance.Advancing research and best practices.By understanding the developmental spectrum, we can better support individuals with developmental challenges and create a more inclusive society.中文回答：发育谱系。

孔东梅采访

Final exam of BIEInterview style: Life histories Lead in---Hello everyone, Here is “Figure Today”. I’m Gongxue. Today our guest is really a legendary figure. When you first meet her, you may have the same feeling like me-----very familiar with her face. Then when I told you she is from the most famous family in china, you may come to understand suddenly. Yes, she is Kong Dongmei, the granddaughter of Chairman Mao, Dongmei, welcome.kong: hello, audiences.Body of interview-----1：Reporter: As we know, you were born in 70s last century, and your childhood wasn’t stand with your parents. How about it 上世纪70年代出生的你，童年并没有在父母身边度过。

那是一个怎样的童年？Kong: I was born in Shanghai, my grandmother and me were living in Shanghai Hunan Road 262, now is Hunan Road Villa. My Grandmother was in a poor health; my parents are in Beijing, my childhood spent with my grandmother and the staff there.The circumstances are exceptional; the grandmother did after the liberation of Beijing until you smash the "gang of four, the restore status of her members. At the time, my parents are motivated to protect my mentality, told me I cannot readily be put other kids to take home. Therefore, I was feeling very lonely at that time. I want to play with the children outside我在上海出生，那时候我和外婆住在上海湖南路262号，就是现在的湖南路别墅。

2018届嘉定区高考英语二模试卷和参考答案(不含听力)

2018嘉定区高考英语二模II. Grammar and VocabularySection AStephen Hawking: Science’s Brightest StarHis family released a statement in the early hours of Wednesday morning confirming his death at his home in Cambridge.Hawking’s children, Lucy, Robert and Tim, said in a statement: “We are deeply saddened that our beloved father passed away today. He was a great scientist and an extraordinary man (21)______ work and legacy will live on for many years.”For fellow scientists and loved ones, it was Hawking’s intui tion and wicked sense of humor (22)______ marked him out as much as the fierce intellect that, coupled with his illness, came to symbolize (23)______ unbounded possibilities of the human mind.Hawking was driven to Wagner, but not the bottle, when he (24)______ (diagnose) with motor neurone disease in 1963 at the age of 21. Doctors expected him (25)______ (live) for only two more years. But Hawking had a form of the disease that progressed more slowly than usual. He survived for more than half a century.Hawking once estimated he worked only 1,000 hours during his three undergraduate years at Oxford. In his finals, he came close (26)______ a first- and second-class degree. (27)______ (convince) that he was seen as a difficult student, he told his examiners that if they gave him a first he would move to Cambridge to pursue his phD. Award a second and he threatened to stay. They opted for a first.Those who live in the shadow of death are often those who live most. For Hawking, the early diagnosis of his terminal disease, and (28)______ (witness) the death from leukemia of a boy he knew in hospital, aroused a fresh sense of purpose. “(29)______ there was a cloud hanging over my future, I found, to my surprise, that I was enjoying life in the present more than before. I began to make progress with my research,” he once said. Taking up his career in earnest, he declared: “My goal is simple. It is a complete understanding of the universe, why it is (30)______ it is and why it exists at all.”He is kindlyThe other evening at a dancing club a young man introduced me to Mr. and Mrs. F. Scott Fitzgerald, and Scott seemed to have changed a lot from the first time I met him at Princeton, when he was an eager undergraduate trying his best to __31__ himself into a great author. He is still trying hard to be a great author. He is at work now on a novel which his wife __32__ me is far better than This Side of Paradise, but like most of our younger novelists he finds it __33__ to produce a certain number of short stories to make the wheels go around. That The V egetable, his play, did not receive a Manhattan presentation seems to have disappointed rather than discouragedhim. He is still __34__ light-hearted.I have always considered him the most brilliant of our younger novelists. Not one of them can tough his style, nor the superb quality of his satire(讽刺). He has yet to put them in a novel with carefulness of conception and __35__ of character. He can become almost any kind of writer that his peculiarly restless character will __36__.Born in St. Paul, he attended Princeton, served in the Army, wrote his first novel in a training camp, achieved fame and fortune, married a Southern girl, has a child and lives in New York. At heart, he is one of the kindliest of the younger writers. Artistry means a great deal to F. Scott Fizgerald, and into his own best work he __37__ great efforts. He demands this in the work of others, and when he does not find it he criticizes with passionate earnestness. I have known him, after reading a young fellow-novelist’s book, to take what must have been hours of time to write him a lengthy, careful __38__.Just what he will write in the future remains __39__. With a firmer reputation than that of the other young people, he yet seems to me to have achieved rather less than Robert Nathan and rather more than Stephen Vincent Benet, Cyril Hume. His coming novel should mean a definite prediction for future work. It is to be hoped that from it will be __40__ the seemingly unavoidable modern girls.III. Reading ComprehensionSection AStandards for Schools: Developing Organizational Accountability(绩效) Quality teaching depends on not just teacher’s knowledge and skills but on the environment in which they work. Schools need to offer a coherent curriculum focused on higher-order thinking and performance across subject areas and grades, time for teachers to work __41__ with students to accomplish challenging goals, opportunities for teachers to plan with and learn from one another, and regular occasions to evaluate the outcomes of their __42__.If schools are to become more responsible, they must, like other professional organizations, make evaluation and assessment part of their everyday lives. Just as hospitals have standing committees of staff that meet regularly to look at evaluation data and discuss the __43__ of each aspect of their work – a practice reinforced by their accreditation(评定) requirements, - schools must have regular occasions to examine their practice and effectiveness.As Richard Rothstein and colleagues describe in Grading Education: Getting Accountability Right, school-level accountability can be supported by school __44__, like those common in many other nations, in which trained experts evaluate schools by spending several days visiting classrooms, __45__ samples of student work, and interviewing students about their understanding and their experiences, __46__ looking at objective data such as test scores, graduation rates, and so on. In some cases, principals accompany the inspectors into classrooms and are asked for their own evaluations of the lessons. In this way, the inspectors are able to make __47__ about the instructional and supervisory competence(能力) for principals. As described earlier, inspectors may also play a role in ensuring the __48__ and comparability of school-based assessments (as in England and A ustralia), as well as school’s internal assessment and evaluation process (as in Hong Kong).In most countries’ inspection systems, schools are rated on the quality of instruction and otherservices and supports, as well as students’ __49__ and progress o n a wide range of aspects, including and going beyond academic subject areas, such as extra-curricular, personal and social __50__, the acquisition of workplace skills and the __51__ to which students are encouraged to adopt safe practices and a __52__ lifestyle. Schools are rated as to whether they pass inspection, need modest improvements, or require serious intervention(介入), and they receive extensive feedback on what the inspectors both saw and __53__. Reports are publicly posted. Schools requiring intervention are then given more expert __54__ and support, and are placed on a more frequent schedule of visits. Those that persistently fail to pass may be placed under local government control and could be __55__ if they are not improved.41. A. occasionally B. closely C. strictly D. peacefully42. A. challenges B. competence C. curriculum D. practices43. A. effectiveness B. faults C. progress D. requirements44. A. instruction B. protection C. inspection D. consideration45. A. taking B. improving C. examining D. copying46. A. as far as B. rather than C. other than D. as well as47. A. judgments B. decisions C. inquiries D. suggestions48. A. quantity B. quality C. instruction D. support49. A. education B. performance C. attention D. interest50. A. responsibility B. structure C. resources D. benefits51. A. frequency B. consistence C. satisfaction D. extent52. A. comparable B. healthy C. different D. unique53. A. appreciated B. criticized C. recommended D. rewarded54. A. attention B. programs C. evaluation D. explanations55. A. set down B. put down C. closed down D. pulled downSection B(A)Eye Scan Technology Comes to SchoolsABC News: Parents who want to pick up their kids at school in one New Jersey district now can submit to iris(虹膜) scans, as the technology that helps keep our nation’s airports and hotels safe begins to make its way further into American lives.this high-tech security system on Monday with funding fromthe Departmen t of Justice as part of a study on the system’seffectiveness.As many as four adults can be authorized to pickup each child in the district, but in order to be authorized tocome into school, they will be asked to register with the district’s iris reco gnition security and visitor management system. At this point, the New Jersey program is not a must.If someone tries to slip in behind an authorized person, the system causes an alarm and redflashing lights in the front office. The entire process takes just seconds.This kind of technology is already at work in airports around the country like Orlando International Airport, where the program has been in operation since July. It has 12,000 subscribers who pay $79.95 for the convenience of submitting to iris scans rather than going through lengthy security checks.An iris scan is said to be more accurate than a fingerprint because it records 240 unique details—far more than the seven to twenty-four details that are analyzed in fingerprints. The chances of being misidentified by an iris scan are about one in 1.2 million and just one in 1.44 trillion if you scan both eyes.Phil Meara, the Freehold District official, said that although it was expensive, the program would help schools across the country move into a new frontier in child protection. “This is all part of a larger emphasis, here in New Jersey, on school safety,” he said. “We chose this school because we were looking for a typical slightly urban school to launch the system.”Meara applied for a $369,000 grant on behalf of the school district and had the eye scanners installed in two grammar schools and one middle school. So far, 300 of the nearly 1,500 individuals available to pick up a student from school have registered for the eye scan system.56. Why does the Freehold Borough School District adopt the eye scan security system?A. To ensure the school safety and efficiency of picking up children.B. To encourage more students to register in New Jersey urban schools.C. To test the effectiveness of school security and management system.D. To collect the information of the children and their beloved parents.57. What makes the eye san system more accurate than the fingerprint system?A. Processing the data of the authorized people faster.B. Identifying the data of the adults to pick up children.C. Submitting the data of the authorized people conveniently.D. Providing far more unique details of the authorized ones.58. How does Phil Meara help to protect the safety of children?A. By asking people to register with the security system.B. By applying for grant to install eye scanners in schools.C. By asking the department of justice to fund this program.D. By turning to Orlando International Airport for help.59. The eye scan system can be best described as ______.A. safe and cheapB. portable and usefulC. smart and accurateD. popular and helpful(B)Senior Manager Major Gift Fundraising & Special Projects Blind Veterans UK is the national charity helping blind ex-service men and women lead independent and fulfilling lives. We offer blind veterans access to the highest quality of services to help them discover life beyond sight loss. We have an exciting opportunity for an innovative and resourceful individual to join our Partnerships team based at our headquarters in London. The team focuses on securing donations from HNWIs, Trusts and Companies. This role focuses on securing support from HNWIs. The special projects aspect of the role relates to annual activitiesthat offer an opportunity to develop relationships with the target audience.We are looking for an experienced individual with a sound track record in the following areas:●Identifying prospects with the capacity and tendency to support●Developing and implementing cultivation and marketing strategies●Managing a document of current as well as prospective major donors●Planning and driving peer to peer fundraising●Organizing promotion events●Delivering against a personal target and team targetsThe successful candidate will also have some people management experience and an expert in major gift fundraising processes will be considered as priority.In return for your talent, we offer competitive conditions of service and a conducive environment. To apply, please send your up to date CV and Supporting Statement of not more than 500 words to Recruitment.Ldn@, outlining how your skills and experience meet the person specification.Interview date: Week starting from 26 March 2018Please note only applicants who submit a CV with a supporting statement will be considered.Due to the high number of enquiries and applications we receive for our vacancies we don’t acknowledge each one – if you hav en’t heard from us within a week of the closing date, please assume that we won’t be inviting you for an interview. You are, of course, welcome to try again if a suitable post comes up. We are unable to provide feedback to candidates not shortlisted for interview.60. The passage is mainly written to ______.A. invite people to join the fundraising eventsB. seek the right person to be Senior ManagerC. inform the blind veterans of money serviceD. attract the interest of potential donors61. According to the passage, which of the following statement is TRUE?A. The application fails if one isn’t informed before 26 March.B. The application should include a lengthy personal statement.C. All the applicants will receive an invitation before interview.D. The applicants should send his application when he is free.62. What experience is most likely to help a candidate stand out?A. People management experience.B. Annul activities experience.C. Peer to peer fundraising experience.D. Large-scale fundraising experience.(C)As businesses and governments have struggled to understand the so-called millennials—born between roughly 1980 and 2000—one frequent conclusion has been that they have a unique love of cities. A deep-seated preference for night life and subways, the thinking goes, has driven the prosperity of urban cores across the U.S. over the last decade-plus.But there’s mounting evidence that millennials’ love of cities was only a passing fling(放纵). Millennials don’t love cities any more t han previous generations.The latest argument comes from Dowell Myers, an urban planning professor at USC. As they age, says Myers, millennials’ presence in cities, will “be evaporating…through our fingers, if we don’t make some plans now.” That’s because millennials’ preference for cities will fade as they start families and become more established in their careers.It’s about more than aging, though. Demographer William Frey has been arguing for years that millennials have become ‘stuck’ in cities by th e 2008 downturn and the following slow recovery, with poor job prospects and declining wages making it harder for them to afford to buy homes in suburbia.Myers, too, says observers have confused young people’s presence in cities with a preference for cities. Survey data shows that more millennials would like to be living in the suburbs than actually are. But the normal career and family cycles moving young people from cities into suburban houses have become, in Myers’ words, “a plugged up drain.”But unemployment has finally returned to healthy lows (though participation rates and wages are still largely depressing), which Myers says should finally increase mobility for millennials.Other trends among millennials, supposedly matters of lifestyle preference, have already turned out to have been driven mostly by economics. What was once considered their broad preference for public transit may have always been a now-reversing inability to afford cars. Even decades-long trends towards marrying later have been stressed as today’s young people struggle for financial stability.Investors are already taking the idea that millennials will return to old behavior patterns seriously, putting more money into auto manufacturers and developers. But urban lifestyles, up to and including trendy bars, aren’t just modern—they’re a part of what powers a city’s economic engines, bringing people together to explore new ideas, create companies, and build careers.From the 1960s to the 1990s, we saw that suburbanization also means an economic and social hollowing out for cities. Now that the economic restrictions are coming off today’s young city residents, cities that want to stay vibrant have to figure out how to convince them—and their growing families—to stick around.63. Over the last decade, what is thought to have ensured the prosperity of the city?A. Fast economic development.B. Around-the-clock club services.C. Convenient public transport.D. Well-established careers.64. Why are Millennials about to leave city?A. It is too expensive for them to buy apartment in cities.B. They find it difficult for to seek a good job in cities.C. It is easier to get married moving to the suburban.D. They are more confident with their economic situation.65. What does th e author mean quoting Myer’s “a plugged up drain”(para 5)?A. Millennials are reluctant to leave attractive cities.B. Millennials are stopped from moving to the suburbs.C. Millennials are unwilling to be cut off from the suburban.D. Millennials are afraid of another economic decline.66. How does the author feel about the suburbanization?A. sign of stable finance.B. A growth of health issues.C. A conflict of new ideas.D. A loss of modern life.The Minoans: A Forgotten PeopleThe first advanced culture in ancient Greece was the Minoan culture. For thousands of years, knowledge of these people survived only in Greek myths. In the late 19th century, archaeologists began to unearth ruins. This inspired Arthur Evans to begin digging on the island of Crete near mainland Greece. On a dig in Kbossos, Evans found an ancient palace. Experts think that it was the palace of King Minos, a central figure in many Greek myths.____67____ With his team, he uncovered a vast structure, varied works of art, and many hieroglyphic records, These finds, together with later finds, comprise all that experts know about Minoan culture.From the evidence experts gathered, it is clear that the Minoans were ahead of their time. The palace at Knossos was five floors high with hundreds of rooms. Buildings throughout the ancient city had plumbing and flush toilets. Stone pavement lined the surfaces of the roads. In addition, the Minoans possessed a highly developed naval fleet for long-distance trade. ____68_____ These records confirm the central role of commerce in culture.Expert analysis of the evidence also offers insight into some aspects of Minoan society. ____69____ Ruins and artwork suggest that people of all classes enjoyed a high degree of social and gender equality. Religious icons show that Minoans worshiped bulls, the natural world, and many female gods.An unusual feature of Minoans culture was the pursuit of leisure interests. Sport and visual arts were central to Minoan life. Boxing and bull jumping, a sport in which players jumped over live bulls, were popular. Although bull jumping may have served some ritual purpose, experts believe that it was done mostly for fun. Similarly, although some works of art showed political and religious themes, other works served only as pleasant décor(装饰品). ____70_____ The Minoans met their demise after a series of natural disasters. Experts believe that groupfrom the Greek mainland capitalized on these events and looked over the island.IV. Summary WritingThe Conflict of the OrdersThe types of people who served as officials in the Roman government changed over time. These changes stemmed from the attempts of common people to more rights. The struggles became known as the Conflict of the Orders.In the early republic, Romans were divided into two classes of people: patricians and plebeians. Patricians were powerful landowners who controlled the government. As nobles, they inherited their power. Plebeians, who made up most of the population, were mainly farmers and workers. For many years, plebeians had few rights. They could vote, but they were barred from holding most public offices. Plebeians could not even know Roman laws because laws were not written down. In court, a judge stated and applied the law, but only patricians served as judges.Over time, plebeians increased their power through demand and strikes. They gained the right to join the army, hold government office, form their own assembly, and elect leaders. In one of their greatest victories, they forced the government to write down the laws of the Roman Republic. In about 450, B.C. the Romans engraved their laws on tablets called the Twelve Tables. The laws were placed in the Forum, the chief public square, for all to view.The first plebeians were appointed to the government in the late 400s B.C. After 342 B.C., a plebeian always held one of the consul positions. By about 300 B.C. many plebeians had become so powerful and wealthy themselves that they joined with patricians to form the Roman nobility. From that time on, the distinction between patricians and plebeians was not a important. Membership in the nobility was still very important, however, since government officials were not paid a salary, only wealthy nobles could afford to hold office. Thus, the nobles still controlled the republic.V. Translation72. 他在会议上提出的建议值得三思。

考研作文学生补充

一满分句型：一there be1 There is strong evidence to show that2 There is a growing worldwide awareness around the world that3 There is absolutely no reason for us to believe that a brighter future for the world is an impossibility4 There is little doubt that5 There are those who claim that二从句1. 主语从句:It is quite understandable that one is keen on these who are handsome and fashionable.It really does not matter whether a cat is black or white, but instead whether or not it can catch the mouse.What is agreed upon the old man is that they should be respected and supported.2. 同位语从句1) The solution must be found that…2) The reason is quite obvious why many youngsters pursue their idols crazily.3) Environmental pollution, a problem threatening human’s existence, has become increasingly serious4) The enormous net ,which symbolizes the internet , has connected together everyone ---young and old, men and women.3.定语从句非限制性定语从句修饰前一句话1) Which leaves us a deep impression2) Which provokes the public’s widespread concerns3) Which brings us the unnecessary/unexpected trouble4 ) which gives rise to dreadful consequences.4.状语从句1）Although immediate solutions for solving problems surrounding the old remain elusive, public recognition of the necessity to respect them will represent the first step in finding effective solutions.2）Although there is a general belief among the young that pursuing their favorite stars is quite acceptable, I , so far, beg to disagree描述图表3）A young man , sitting in a barber’s shop, spends 300yuan ,a high price for many Chinese,having becham’s hair style , which provokes the public’s widespread concerns四特殊结构一）插入语：插在主谓中间，用逗号或破折号隔开。

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The conception of mathematics among Hong Kong students and teachers

Ngai-Ying Wong The Chinese University of Hong Kong, Hong Kong nywong@cuhk.edu.hk

Abstract Though Hong Kong students outperformed their Western counterparts in international comparisons in computational problems, their performance in non-routine mathematics questions is not that brilliant. This could be related to the conceptions of mathematics both among the students and among the teachers. When mathematics is regarded as an absolute truth or a set or rules for playing around with symbols, students would tend to treat doing mathematics as the memorisation of algorithms and learning mathematics as a process of transmission. With the use of situations similar to those in Kouba and McDonald (1991), we found that students possess a relative narrow conception of mathematics. Later, we proceeded to make more in depth investigations by the use of open-ended questions. Again, it was found that students did not perform well with open-ended questions and in general, they tried to approach a mathematical problem by searching a rule that fits by identifying what is given, what is being asked and the topic the problem belongs. Not only that, evidence has shown that such a way to approach mathematical problem is largely shaped by the way they are experiencing learning, task demands, and classroom environment. In other words, such a narrow conception of mathematics which exist both within the students and in the classroom culture has led to students tackling mathematical problems by the search of rules rather than via a conceptual understanding of the context.

1. How we launched the endeavour The research team (other members: Chi-Chung Lam and Ka-Ming Wong) started off the investigation as a curriculum issue. Students’ conception of mathematics was taken as part of the “attained curriculum” (Travers & Westbury, 1989). First of all, Asian students outperformed their Western counterparts in international comparisons in mathematics, (Cai, 1995; Leung & Wong, 1996, 1997; Stevenson & Lee, 1990), but we suspected that they may lack conceptual understanding. As one facet of it, we suspected that students only possess a narrow conception of mathematics. Secondly, the mathematics curriculum was under reform in which the author was actively involved. As a first step, we need to make a full situational appraisal, to identify the strength and weaknesses of the current curriculum. There, students’ conception of mathematics was again one of the foci of investigation (Wong et al, 1999). Thirdly, we made some tried out the use of history mathematics in teaching (Lit & Siu, 1998). To evaluate the effectiveness of such a curriculum, we saw that conventional tests do not suffice. We incorporated assessments in the affective domain, including beliefs. All these led us to a more in depth investigation on the conceptions of mathematics. At the same time, we realised that a well-established methodology for such a series of studies is in need.

2. Our project Initially we targeted at the development of a methodology, an instrument, that could be used in future research. As we moved on, why Hong Kong students possess such a narrow conception of mathematics become prominent. In brief, it was shaped by the long time indulgence of a learning environment that lacks variation. It was thus a natural step forward to study the conception of mathematics among teachers.

(a) Prologue (1992-93): use of open-ended questions to tap when did students regard themselves as having understood some mathematics (Wong, 1993, 1995).

(b) Phase 1 (1996-97): use of hypothetical situation and asked student to judge whether it is “doing mathematics” in each case (Wong, Lam, & Wong, 1998).

(b') Test trail of results obtained in phase 1 (1997-98): testing of reliability of a questionnaire developed from the results obtained in phase 1 (Wong et al, 1999).

(c) Phase 2 (1997-98): use of open-ended mathematics problems to tap students’ approaches to tackling these problems in relation with their conceptions of mathematics (Wong, Marton, Wong, & Lam, in preparation).

(d) Phase 3 (1998-99): investigation of teachers’ conceptions of mathematics and mathematics teaching by questionnaires and by interviews (part of the result can be found in Wong et al, 1999).

3. Students’ conceptions of mathematics 3.1. When did students regard themselves as having understood some mathematics ? Two hundred and forty one Grade Nine students in Hong Kong were invited to respond to the open-ended questions on approaches towards mathematics problems: (a) “What would you do first in facing a mathematics problem ? (b) “What methods do you usually use in solving mathematics problems ?” (c) “What is the essential element in successful mathematics problem solving ?” (d) “Which do you think is the most important step of a successfully solved mathematics problem ?” and (e) “If you were to score a completed mathematics problem, which do you think is the most important part ?”. It was found that “trying to understand”, “revise and work hard” and “asking others for help”.

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