The Mathematics Classroom in Beijing, Hong Kong and London Author(s): F. K. S. Leung Reviewed work(s): Source: Educational Studies in Mathematics, Vol. 29, No. 4 (Dec., 1995), pp. 297-325 Published by: Springer Stable URL: http://www.jstor.org/stable/3482673 . Accessed: 18/08/2012 08:23 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp
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THE MATHEMATICSCLASSROOMIN BEIJING,HONG KONG AND LONDON
ABSTRACT.This paperreportsa classroom observationstudy which intends to characterise the instructionalpractices in junior secondarymathematicsclassrooms in Beijing, Hong Kong and London, focusing on the differentculturalbeliefs pertainingto mathematics and mathematicsteaching and learningbetween the Chinese and Westerncultures. The results show that there are striking differences in classroom practicesbetween the three places, and the differencesseem to be relatedto the differencesin attitudestowards mathematicsand mathematicsteachingand learning.The findingspoint to the potentialof the culturalperspectivein interpretingresultsof comparativecurriculumstudies.
Researchersin comparativecurriculumstudies are increasinglyaware of the need to look at instructionalpracticesin the classroomas a characterisation of the curriculumand as the basis for interpretingdata on student outcomes. The IEA Second InternationalMathematicsStudy (SIMS) and ThirdInternationalMathematicsandScience Study(TIMSS),for example, depictthe curriculumas consistingof threelevels, the intented,implemented and attainedlevels (Traversand Westbury,1989, pp. 5-8; Robitaille, 1991, pp. 4-6), and stress the "need for extensive and detailed information on the teaching of mathematics"(Traversand Westbury,1989, p. 5). However, in studyinginstructionalpractice,many comparativestudies rely on studentreportor teacherself-reportof the teachingprocess(SIMS, TIMSS;Chang, 1984) and very few studiesincludea componentof classroom observation.But "to critiqueor appraisethe operationalcurriculum requiresone to be in a position to observe what classroomactivities actually unfold"(Eisner, 1985:47), and no matterhow well the questionnaires concerning instructionalpractices were set, the data collected could not substitutefor the informationgatheredthroughclassroomobservation. Among the few comparativestudies on the mathematicscurriculum that include a componentof classroomobservationin the past are a series of studies based at the Centre for Human Growth and Development of the University of Michigan (see for example Stigler and Perry, 1988b; EducationalStudies in Mathematics 29: 297-325, 1995. ? 1995 KluwerAcademicPublishers. Printedin Belgium.
Stevenson and Stigler, 1992). Stevenson (1987), in discussing the results of the classroompracticecomponentof the study,reportedthattherewere markeddifferencesin practicebetween the Asian countriesof Taiwanand Japanin the study and the US. For example, the study found that in Taiwanese and Japaneseclassrooms,more time was devoted to mathematics teaching, there was more direct instructionduring mathematicslessons, and teachers were requiredto be in charge of the classroom for fewer hourswhen comparedwith the Americanclassrooms.Stevenson suggested that many of the differences might be reflectionsof the differencesin the culturalbeliefs in the countriesconcerned. This study intends to characterisethe instructionalpracticesin junior secondary mathematicsclassrooms in Beijing, Hong Kong and London based on classroom observations in the three cities. Different cultural beliefs pertainingto mathematicsand mathematicsteachingand learning between the Chinese and Westernculturesfrom the literaturewere identified andused as the focus for the classroomobservation,andattemptswere made to relatethe differencesin classroompracticeto these differencesin beliefs. The choice of the three places for study is significant.Hong Kong has been a British colony for more than one and a half centuries.However, because of her origin from and proximity with China, Hong Kong has never lost her culturallink with the motherland.On the contrary,being the most homogeneously ethnic Chinese communityoutside MainlandChina and Taiwan,most of the Chinese traditionsand values are still retained. Hong Kong's dramaticprosperityunderthe Britishruleandthe consequent development into one of the world's majorcommunicationand financial centreshave howevermadeit a city proneto influenceby Westerncultures. It is this particularculturallocation of Hong Kong with respect to both China and England that allows us to examine the possible influence of culturalbeliefs on classroompractice.
THE CHINESECULTUREIN CONTRASTWITHWESTERNCULTURES
A review of the relevantliteratureshowedthatthe following characteristics of the Chinese culturewere used to explaindifferentpracticesin education and mathematicslearningbetween the Chinese and the Westerners. 2.1. The 'Social Orientation'of the Chinese Comparedwith Westerncultureswhich stressindependenceand individualism (Taylor,1987:235), the Chineseemphasizeintegrationandharmony (Sun, 1983). Yang(1981:161) uses the term'social orientation'(as opposed
to individual orientation)to label this nationalcharacterof the Chinese. Relatedwith this social orientationof the Chinese are characteristicssuch as compliance (Bond and Huang, 1986; Lin, 1988), obedience, respect for superiors and filial piety (Liu, 1986). These characteristicsperhaps explain the relatively good discipline of studentsobservedin classrooms influencedby Chinese culture (Waresand Becker, 1983; Leestma et al., 1987), but they were also used by researchersto explain the rigidity of thought and the lack of originality in Chinese and Hong Kong students (Liu and Hsu, 1974; Douglas and Wong, 1977). 2.2. The ChineseStress on Memorizationand Practice The Chinese were known to place strongemphasison memorization(Liu, 1984, 1986) andpractice(Liu, 1986;Stevensonet al., cited in Cunningham, 1984), and researchershave reportedon the seemingly superiorperformance of Chinese subjectsin memorizationtasks (Hoosain, 1979; Huang andLiu, 1978). This stress on memorizationanddifferencesin memorization ability may have significant implicationsfor mathematicsteaching and learning. 2.3. TheHigh Expectationson StudentAchievementand the Attribution of Success and Failure Chinese parentsand teachersare known to attachgreatimportanceto the education and achievement of their children and students(Sollenberger, 1968;Ho andKang, 1984). One manifestationof suchhighexpectationsare the demandingmathematicstextbooksfound in Chinaand Japan(Stevenson, 1985; Stigler and Perry,1988a;WaresandBecker, 1983). These high expectationsmay be relatedto the ChineseandJapaneseattributionof academic successes andfailures.Chineseparentsandteachersaremore likely to attributetheirchildren'ssuccess and failureto controllablefactors(e.g. effort) ratherthan uncontrollableones (e.g. ability) (Weineret al., 1971; Stevenson et al., 1986; Hess et al., 1987; Stevenson, 1987). 2.4. TheAttitudeTowardsStudy The traditionalChinese attitudetowards studying and learning is that it is a hardship.One should pay the price of diligence and perseverancein order to succeed, and is not supposed to "enjoy"the studying (Garvey and Jackson, 1975; Hess et al., 1987). As Stevenson (1987) commented, "Asian parentsteach their childrenearly that the route to success lies in hardwork". In this study,the differentbeliefs and attitudesidentifiedabove will be used as the focus for the classroomobservations.
F.K.S.LEUNG 3. METHODOLOGY
Preliminaryvisits to mathematicsclassrooms in three to four schools in each of Beijing, Hong Kong andLondonwere conductedbetweenNovember 1987 and June 1988. Fromthe resultsof the preliminaryvisits and the literaturereview, a simple observationschedule which recordsthe durations of classroom activities and the frequenciesof occurrenceof events under a list of 'sensitizing concepts' that reflect differentviews of mathematics and mathematicsteaching and learning was developed, and the schedule was piloted between December 1988 and March 1989 in four schools in each of the threecities. The resultof the pilot studyshowedthata rigidschedulewas not able to capturethe diversityof practicesin the threeplaces. So it was decidedthat for the main study, in additionto using an observationschedule (slightly revised) to recordthe durationsof classroomactivitiesandthe frequencies of occurrenceof events underthe list of 'sensitizingconcepts', the events themselves would be recordedin detail, and the lessons would be audiotaped. Detailed narrativefield-notes on non-verbalaspects of the lessons would also be taken. In the schedule, classroom activities include: lecturing, questioning, class-work on an individualbasis and on a group basis, maintainingdiscipline, giving administrativeinstructions,and silence. The list of sensitizing concepts which reflect differentviews of mathematicsand mathematics teaching and learning includes: the use of rigorousmathematical languages;alternativemethodsof solving the sameproblem;trialanderror methodof solution;investigationsor exploration;conformityto teacher's instruction;memorizationof mathematicalresults or processes; expectation on studentsto work hard;and studying mathematicsfor the sake of examinations. The classroomobservationsfor the mainstudywere done in December 1989 for Beijing, Marchto May 1990 for Hong Kong, andNovember1990 to February1991 for London. Six regions in each city were chosen, and three schools were visited in each region, making a total of 18 schools for each city. The 18 schools were chosen from a variety of background in each of the three cities and altogether112 lessons were observed.The distributionof the gradesof lessons observedin the threeplaces is shown in TableI below. At the data-analysisstage, the field notes takenduringclassroomobservations were expandedby going over the audio-recordingsof the lessons. Details missed out in the notes werefilledin, andany incidentsfallingunder the sensitizingconcepts were also noted at this stage. The end resultswere more detailed writtenrecordsof lessons thanthe originalfield notes.
TABLEI Distributionof gradesof lessons observedin the main study China Hong Kong London
9 13 16
Next, from these expanded field-notes, the original categories were amendedso thatthey reflectedwhatwere capturedby the classroomobservations in the three places. Repeated happeningsof strikingevents that were common to the three places or that differed greatly from place to place but which were not included in the original sensitizing concepts were noted, and new categories were constructedso that they accommodatedthese incidents. At this stage, it was decided that data be treatedin two differentways according to their nature:either the durationsof the events were noted, or the occurrenceor otherwise of the events in a lesson was noted.1For example, the amountof teachertalk was noted as a durationwhile mention by the teacherof examinationswould be countedas an occurrence. Lastly, the audio-recordingswere gone over again to note down the durationsor countthe occurrencesof the incidentsfor analysis.In reporting the results of the analyses, the descriptionsof activities in the field notes were available and the relevant parts of the audio-recordingscould be traced when illustrationswere needed. The events or incidentsunderthe two differenttreatmentsare listed in AppendixI.
The schools in these threecities differedtremendouslyin theirphysicalsetting, facilities and structure.The most obvious differencewas the number of studentsin each class. The figures in TableII below show the number of studentsin each class observedfor the threecities. All the schools in Beijing andHong Kong adoptedwhole-class instruction, with studentssittingin rows facing the teacherandthe teacherleading all the activities in the classroom.In London,24 of the 40 lessons observed also followed this mode of instruction,though sometimes studentssat in groupsinsteadof sitting in rows. The remaining16 lessons followed some individualizedlearningprogrammes,and nearly all of these lessons were
TABLEII Numberof studentsin a class Beijing Hong Kong Mean 42.8 38.7 Maximum 54 43 Minimum 30 21 Standarddeviation 6.4 4.8
London 22.2 30 9 4.8
conductedwith studentssitting in groups.These differencesin class size and instructionalmode in the threeplaces should be borne in mind in the following analyses and discussions. 4.1. TheFlow of EventsDuring the Lesson The flow of events in the lessons of the threeplaces will be describedand compared based on the data on the durationsof various events and the occurrencesof events relatedto classroompractice.AppendixII displays the mean, standarddeviationand inter-quartilerangeof the percentagesof time spent on various activities, and AppendixIII shows the occurrences of the variousevents in the classroom. Briefly speaking, a typical Beijing lesson would start by the teacher revising with studentswork they had learnedin previouslessons, usually done throughquestioning.Then the teacherwould introducethe topic of the lesson (this happenedin 34 out of the 36 lessons observed,andusually the topic was writtenon the board)and startdevelopingthe topic in some length throughexplanation, sometimes incorporatingquestioningin the explanation. One or more examples, usually taken from the textbook, would then be worked out on the board and discussed. From time to time, the contentcovered would be summarized(22 lessons), and students would be asked to attemptsome class-work.Towardsthe end of the classwork time, the teacherwould usually ask a few studentsto come out and write their work on the board.The students'work on the boardwould be discussed and the lesson would be summarizedagain. Lastly the lesson would end by teacherassigning homework. In a typical Hong Kong lesson, the teacherwould startby reminding the studentsof what they had learnedin the last lesson. Then the teacher would go on to explain new content and work examples on the board, sometimes referringstudentsto the textbook (24 out of the 36 lessons). After that class-work would be assigned, and sometimes studentswould be asked to write their work on the board,like the practicein the Beijing
classroom. Then the class-workwould be discussed and the lesson would end by teacherassigning homework. The flow of events in a London lesson thatfollowed an individualized programmewas simple. The lesson would startby teacher dealing with some administrativematters such as roll-call and making sure that the students got the requiredmaterialsfor their work (this happenedin all the 16 lessons observed). Then studentswould work on their own, and the teacherwould go aroundthe room busily helping studentswith their mathematicsas well as recordingthe grades of students'work. At about five minutesbeforethe bell, the teacherwouldask studentsto "packaway", and the teacherwould begin to check whethermaterialstakenby students duringthe lesson were returnedto the properplaces. When the bell rang, the teacherwould ask studentsto leave. For those lessons thatdid not follow an individualizedprogramme,the flow of events was very similarto thatof the Hong Kong case, except that less time was spent on teachertalk and more time on class-work. 4.2. Discussion A number of things can be noted from the figures in Appendix II, the descriptionsabove and the classroom observationfield notes in the three places: Firstly, the Beijing lessons were found to be much more structured than the lessons in Hong Kong and London,and this structurewas found in nearly all the Beijing lessons observed,no matterwhetherthe lessons took place in a key-point school2 in inner Beijing or a "below average" school in the remote ruralareas. In most lessons, the patterndescribedin the paragraphabove was followed strictly.One of the reasons why the Hong-Kong and London lessons appearedless structuredthanthe Beijing lessons is that sometimes teachersin these two places were found making extemporaneousdecisions in their teaching,eitherin the use of examples or in deciding whatthe next step of the lesson shouldbe. These were found in 12 of the 36 lessons observed in Hong Kong, 8 of the 24 whole-class teachinglessons in London,but none in the Beijing classroom.In contrast, the Beijing teachers seemed thoroughlypreparedfor their lessons. Many of the examples used in the lessons were written on a board prepared beforehand(even thoughsome of these exampleswere takendirectlyfrom the textbook), and it could be seen that each step of the lesson followed anotherin a structuredand sometimesrigid manner. Secondly, groupwork was rarelyobservedin the lessons of any of the threeplaces in this study(0 to 2.97%of the lesson time). The only frequent "group activity" observed was that students sitting next to each other
occasionally discussed the class-workthey were doing. This happenedin all threeplaces but was found more frequentlyin London.3 Thirdly, whole-class instructionwas much more common in Beijing (86.30% of the lesson time) and Hong Kong (72.52%) than in London (42.13% and 0%), whereasLondonstudentsspent substantiallymoretime doing "seat-work"(38.35% and 77.84% in contrastto 13.21%in Beijing and 18.93%in Hong Kong). Although whole-class instructionwas the predominantmode of instrucation in both Beijing and Hong Kong, a closer look at the break-downof the whole-class activities time shows some majordifferences.The Beijing teachersused a relatively high proportionof the time (18.39%) involving studentsthroughquestioning(comparedwith 5.03%for Hong Kong), and Hong Kong teachers used more time (9.44%, comparedwith 6.68% for Beijing teachers)in reviewing previous work every lesson, usually at the beginningof the lesson. As for the kind of teacher-talkin the threeplaces, an examinationof the field notes shows thattherewere differentemphases in boththe contentof the talkandthe mannerin whichthe talkwas conducted. Beijing teachersspent a lot of time on expoundingconceptualissues, Hong Kong teachersused most of the teacher-talktime for demonstrating solutions of mathematicsproblems,while teachersin London spent more time discussing mathematicswith students.The pace of the explanationby teachersin Beijing was in generalmuchslowerthanthatin Hong Kong and London. The teacher would help studentsto probe into the mathematics and students were given more time to think about what the teacher was saying. In contrast,teachersin Hong Kong and Londonusually appeared to be rushingthroughthe contentwhile teaching. In London, much more class time was devotedto seat-work.Moreover the kind of seat-work done in London was usually different from that in either Beijing or Hong Kong. For Beijing and Hong Kong, seat-work was essentially practisingskills just learnedfrom the teacher.In London, for those classes that followed an individualizedprogramme,the seatwork (plus occasional help from the teacher)constitutedthe totalityof the implementedcurriculumfor the students,and studentsin the same class were in general learningdifferentthings. Even for those classes that did not follow an individualizedprogramme,therewas more between-student variationin the work that they were doing comparedwith the lessons in Beijing and Hong Kong, because in many cases studentswere allowed to move on at different paces. Very often, some students were well ahead of others and of what the teacher was explaining duringthe whole-class instructiontime, while some were lagging behind.
Fourthly,a substantialproportionof the class-timein LondonandHong Kong was not used for mathematicsteaching and leaming, with the percentage of "off-task"time in London(16.58%to 20.37%)greaterthanthat in Hong Kong (8.55%). In sharpcontrast,in nearlynone of the lessons in Beijing was off-task time observed. The off-task time in Hong Kong was, without exception, due to the late arrivalof the teacherat the classroom,and so the first 3 or 4 minutes after the bell were usually not used for mathematicsteaching. But after the teacher had arrived, there were very few "off-task"times observed (4 occasions) right till the end of the lesson. In London, teachersusually arrivedat the classroom late as well. But in addition,many classes would end aboutfive minutesbefore the end-of-lessonbell (typicallythe teacher would ask studentsto "packaway" at aboutfive minutesbefore the bell), and even during the lesson, there were occasions when the teacher was engaged in some administrativework while studentswere not workingat all (29 out of the 40 lessons).4 These addedup to give an "off-task"figure in Londonmuch higherthanthose in the othertwo places. In Beijing, it seems that there was no off-task time duringthe lessons at all. Typically,the Chinese teacherwould arriveat the classroomone or two minutes before the lesson, and when the bell rang, the lesson began. The lesson usually lasted rightuntil the 45th minute(and sometimeseven extending for a few minutesbeyond the bell). Fifthly, there were more incidentsof maintainingdiscipline in London thanin Hong Kong andBeijing. Incidentsof teachermaintainingdiscipline were observedin 28 of the 40 lessons observedin London,while the correspondingfiguresfor Hong Kong andBeijing were 6 and 1 respectively. Lastly, it was found thatthe teaching in all threeplaces dependedvery much on the textbooks used, althoughthe way the textbooks were used differedfromplace to place. In Hong Kong,some teachersreferredstudents to the textbooks duringtheir explanationof the mathematicsconcepts or techniques, and many teachers used the Class Practice suggested in the textbook as a learningactivity for students.In Beijing, referringstudents to the textbookwas relativelyrare,butthe contentof most lessons followed the textbook closely. The approachof the teaching was identical to that in the textbook and the examples were in generaltaken directly from the textbook. In London, for those lessons that followed an individualized programme,the textbookor the individualizedlearningmaterialswere the sole determinantof the students'learning.In lessons that were conducted in a whole-class setting, the influenceof the textbookon the teachingwas less obvious. Therewere only two (outof 24 lessons) incidentsof reference to the textbook, and in some lessons, materialswere chosen from sources
TABLEIII Events relatedto views of mathematicsand mathematicsteachingandlearning Beijing (36)
Hong Kong (36)
use of rigorouslanguage
a. b. c.
alternativemethods trial anderror investigations
follow teacher's wording follow fixed steps fixed solution format standardtypes
Events B.1. 2.
b. c. d. 4.
other than the textbook that was supposedto be in use. So otherthan the lessons that were conducted in a whole-class setting in London, all the lessons in the threeplaces were very much influencedby the textbooksin use. 4.3. EventsRelated to Viewsof Mathematicsand MathematicsTeaching and Learning Table III above shows the results of the occurrencesof events underthe 'sensitizingconcepts' thatreflectdifferentviews of mathematicsandmathematics teaching and learning, and examples illustratingthe events are given in the discussions thatfollow.
4.4. Discussion 4.4.1. Use of RigorousMathematicalLanguage One common featurein the Beijing classroomswas the stresson the use of exact, rigorousmathematicallanguage.This was recordedin 15 of the 36 lessons in Beijing but none in Hong Kong andLondon.A strikingexample is reproducedbelow: Example 1 (MB0412) It was a year 2 class and the teacherwas discussing the proof of the following theorem: "Ina triangle,if two of the sides are not equal, then theirrespectiveopposite angles are not equal as well, and the angle opposite the longer side is larger." After discussing with students how to tackle the proof and arrivingat the intermediate conclusion that a line needed to be constructedin the triangle, the teacher asked the studentshow the line should be constructedand nameda studentto answer:
"Constructline segment CD so thatAC is equal to AD."
"What?Which is equal to which? Are you saying this (pointingto AC) is equal to this (pointingto AD)? Where does this (pointingto AD) come from?Which do we have first?Do we have you firstor do we have your fatherfirst?"
(silent) (the rest of the class laughed)
"Canwe say AC is equal to AD? How should we say it?"
S: "ADisequaltoAC." T:
"Yes, AD is equal to AC."
and the lesson continued.
Such is the kind of exact mathematicallanguagethatBeijing teacherswere stressing. 4.4.2. A Flexible Viewof Mathematics 126.96.36.199. EmployingAlternativeMethodsin SolvingtheSameProblem. Occurrences under this theme were observed in 10 out of the 36 Beijing lessons, and only once in each of Hong Kong and London. This result is unexpected,but can be explainedby the fact thatmathematicscontentwas treatedmore thoroughlyand the pace of the lessons was slower in Beijing lessons. Typically these incidents happenedwhen the teacher,after discussing a proof of a theoremor a solutionto a problem,asked studentsto suggest alternativeways of provingor solving thatwere not includedin the textbook (if the alternativeswere includedin the textbookand the teacher just went over them with the students,it was not countedas an occurrence of "alternativemethods").In Hong Hong andLondon,the explanationsby teachersusually seemed to be done in a hurry,and there was in general not much time for deep thoughtsor for explorationof alternativeways in tacklingmathematics. 188.8.131.52. Trial and Error Method of Solution of Problems. Only two occurrencesof encouragementof studentsto use trial and errormethods were observedin Londonandnone in Hong Kong andBeijing. Therewere two more incidents in the London lessons where studentswere observed to be adopting a guessing strategy in solving problems, but these were more wild guesses ratherthan "informedguesses" (see Polya, 1948), and so cannotbe consideredas trial and error.An example of trialand erroris given below (ML0233): It was a year 3 lesson andthe class was doing divisionof numbers,andstudentsencountered a question in the textbook which asks what numbersdivide 4 will give a quotientgreater thanone. The teachersaid to the class: T:
"I suggest you do this by trial and error.Try a number,if it is too small, try a bigger one."
An example of wild guesses is given below: (ML0123) It was a bottom set in the third year. The students were working on differentthings in theirseats, and a studentasked the teacherhow many gramstherewere in a kilogram.The teacherdrew the attentionof the class and asked: T:
"Anybody know how many grams ... shuu
can you listen ... Andrew ... how
many gramsin a kilogram?" (Studentsall shoutedanswersout from their seats.) S 1: "ten." S2: "fifty,twenty,forty." S3: "one hundred." S2: "thirty,thirty-five,ten,..." (Teacherlaughed.) T:
"Comeon, keep going up."
S4: "fifty." T:
S4: "a hundred." T:
S 1: "two hundred." T:
S4: "a thousand." T:
"A thousand,yes, a thousand.What does "kilo" always mean? What does the word "kilo"always tell you?..."
And the lesson continued.
184.108.40.206. Investigationsor Explorations. These occurredin five out of the 24 whole-class instructionlessons in London,but none in Hong Kong and Beijing. Four of the five lessons were formally investigationlessons (the teacherannouncedat the beginningof the lesson thatthe class was to do an investigationin that lesson), the remainingone was a lesson which was devoted totally to exploring patternsand making generalisations.In addition,therewere some activitieswithinthe individualizedprogrammes thatcould be consideredinvestigations.Here are two examples of investigations done in the Londonclassrooms: Example 1: (ML1723) "There are 10 people in a room and they all shake hands with each other. How many hand-shakesare there?"(year 3) After the studentshad explored the cases for 2,3,...,10 people, the teacherintroducedthe term"triangularnumbers"to students.Thenhe askedstudentshow manyhand-shakesthere would be if therewere 100 people, and went on to generalizeto the numberof hand-shakes
when there were n people in the room.
Example 2: (ML1633) "A newspaperboy has to deliver two copies of newspaperto two of five households,but he has forgottenwhich two. How many ways are therefor the two copies to be delivered?" (year 3) The studentswere then askedto explorecases for differentnumbersof copies of newspaper anddifferentnumbersof households,and the teacheragainintroducedthe term"triangular numbers"to students.The teacher did not generalize the investigationfurther(e.g. to n copies and m households)though.
4.4.3. Conformityand a Rigid Viewof Mathematics 220.127.116.11. Teacher Requiring Students to Follow the Teacher's Wording Exactly. This was observed in 9 of the 36 lessons in Beijing lessons but not in any of the Hong Kong or London lessons. It usually occurred when studentswere either answeringthe teacher'squestion or working a problemon the board.The teacher,in respondingto the students'answer or in going over students'workon the board,would "correct"the students' answers or work so that the wording followed that of the teacher's,even thoughthe students'own wordingwas not wrong.Forexample:(MB1022) It was a year 2 class and the teacher was discussing the solution of the following equation: X4- 7x2 - 8 = 0
The teachernamed a studentto offer a solution and wrote it on the board. Solution:Let x2 = y. Then originalequationbecomes 2-
7y - 8 = 0
(y - 8)(y + 1) = 0 y = 8, y = -1. The studentcontinued: S:
"Because x squareis equal to y, thereforex squareis equal to 8 or x squareis equal to..."
"Youbettersay 'When y is equal to 8, x squareis equal to 8'."
Teacherwrote her own version on the board: When y = 9, x2 = 8
x = 2V'
When y =-1,
no real solution
x2 = I-
18.104.22.168. Teacher Requiring Students to Follow Fixed Steps in Solving Problems. This was a more common practicein Beijing (14 occurrences)
thanin Hong Kong (2 occurrences),but was not foundat all in the London classroom. It was meant to help studentsto solve mathematicsproblems systematically.For example, in a year one lesson on applicationsof linear equations in Beijing, the teacher startedthe lesson in the following way (MB0211): T: "Whatarethe steps for solving application[word]problemson [linear]equations?" (pause) "Mary." Mary recited:'The steps in solving applicationproblemsare:one, examine the question; two, set up the unknowns;three,list the equivalentquantities;four,expressusing algebraicexpressions;five, set up the equation;six, solve the equation." T: (to the whole class) "Is thatcorrect?" The whole class: "Yes."
Incidentally,this is also a good example of encouragingstudentsto memorize mathematicsprocesses. 22.214.171.124. TeacherRequiringStudentsto PresentSolutions in a Fixed Format. Related to the last theme were incidentswhen the teacherdictated a formatof solution to mathematicsproblemsfor studentsto follow. This happenedfive times in Beijing and four times in Hong Kong. What happened was the teacherwould write down a workedexample on the board or discuss a workedexample in the textbookthoroughlyand then require studentsto follow the formatwhen they solve thattype of problem.It was meant to help studentspresent mathematicssystematically,but the rigid mannerthis was imposed on studentsmight result in encouragementof conformityin expressing mathematics. 126.96.36.199. TeacherCategorizingProblems into StandardTypes. This was a practice found only in some Beijing lessons (9 occurrences)and not in any of the Hong Kong andLondonclassrooms.Forexample,some Beijing teacherswould classify problemsof applicationsof linearequationsinto a numberof types (mixing liquids, two people or cars travellingat different speeds on the same road,a numberof people doing a piece of worktogether or separatelyetc.), and would deal with each type of applicationsone by one. 4.4.4. Memorizationof MathematicalResultsand Processes. Incidentsof encouragementof memorizationwere observedin 16 lessons in Beijing, two lessons in Hong Kong andnone in London.Thesewereincidents when the teacherasked the studentsspecificallyto memorizesome mathematicalfacts, or when the teacherasked studentsfor the statement of a theoremor some mathematicalresultsand studentswere expected to
recite the results.An example from a yeartwo class in each of Beijing and Hong Kong is given below: Beijing (MB1322) T: "Whatis the theoremfor the propertiesof isosceles trianglesandits two corollaries? ... Susan." Susan(reciteby heart):"Thetheoremfor the propertiesof isosceles triangles:'Thetwo base angles of an isosceles triangleare equal.' Abbreviation,'Equalangles opposite equal sides.' Corollary one: 'The angle bisector of the apex angle of an isosceles trianglebisects and is perpendicularto the base,' Corollarytwo: 'All the angles in an equilateraltriangleare equal, and each angle is equal to sixty degrees." T: "Canyou repeatthe second corollaryagain?" Susan: "All the angles in an equilateraltriangleare equal, and each is equal to sixty degrees." T: "Sit down." (to the whole class) 'The way she put it was not too bad." [This was supposedto be a complimentto Susanfor her answer.]
Hong Kong (MH0412) The teacherwas introducingthe topic of simple and compoundinterest,and he explained the terms "interest","principals","amount"etc. to students.Then the teachersaid to the students: T:
"Now copy these terms into your notebooks.You have to memorizethese terms and their meanings."
Then studentscopied what were on the boardinto their notebooks.After that, the teacher went over the terms again with students.
4.4.5. Expectationon Studentsto WorkHard No incident at all underthis theme was observed.May be expectationon studentsto work hardwas a view not held by teachersin all threeplaces, or may be the expectationwas conveyed to studentsin more subtle ways not detectedby the observationsin this study. 4.4.6. TheInfluenceof Examinations Mention by the teacher of examinationswas recordedin five lessons in Beijing, four lessons in Hong Kong and seven lessons in London. An example from a year threelesson in Londonis given below (ML0723): The lesson was on PythagorasTheorem,andthe studentswere workingon some problems on the theoremin theirseats while the teacherwas going aroundhelping studentswith their work. Then the teacherdrew the attentionof the students:
MATHEMATICS CLASSROOMS T:
"Everyonedraws a triangleout. Get used to drawingtrianglesout because you often can't decide what to do until you saw a drawingof it. And even if you can, you get marksfor showing drawings.The examinerdoesn't know what you are thinking ..."
Mentioningexaminationsis of course only one manifestationof the influence of examinations.Whetherthe contentcovered in the lessons is confined to those found in the examinationsyllabus and whetherthe kind of exercise done in class is devised so thatit is similarto thatencounteredin examinationsare othermanifestations.It was mentionedabove thatclasses in all threeplaces followed the textbooksvery closely. As textbooksin Chinaand Hong Kong were very much influencedby the examinationsyllabus when they were writtenand includedmainly paperand pencil type of exercise while those in England (especially those for individualized programmes)seemed to be less influencedby the examinationsyllabus and contained more activities, it could be arguedthat Beijing and Hong Kong lessons were more influencedby public examinationsif we look at students' activities ratherthan listening to the teacher'swords duringthe lessons. 5. SUMMARY
In summary,the findings from the classroomobservationshow thatthere was much more concern for individualdifferences in London, as manifested by schools adopting individualizedlearningprogrammes,and by teachersspending less time on whole-class instructionand more time on seat-workand allowing studentsto proceed at differentpaces. In Beijing and Hong Kong, studentswere more exposed to direct instructionsfrom the teacher(this is consistentwith the findingby Stevenson (1987) quoted at the beginningof this paper).In Beijing, mathematicslessons were more structuredand there was greater stress on mathematicsconcepts, while in Hong Kong more emphasis was put on practisingmathematicalskills. In London, the stress was on studentsdoing mathematicson their own, and for those lessons thatfollowed an individualizedprogramme,students were interactingwith the textbooksor word-cardsmost of the time. Also, studentsin Beijing were being askedmorequestionsby the teacherandhad more opportunitiesto express mathematicsverballyin front of the whole class. They also spent less time during lessons on off-task activities. To put it crudely,it can be said thatBeijing teachersemphasizedthe content or concepts of mathematics,Hong Kong teachersemphasizedmathematical skills, while London teachersemphasizedexperiencingand enjoying mathematics.
From the discussions above it can also be seen that the classroom practice of Beijing teachers reflected a more rigid view of mathematics thanthe practicesin Hong Kong andLondon.Thereis a greatemphasison the use of rigorousmathematicallanguage,while investigationalactivities and the method of trial and errorwere not common at all. There is also a much greater stress on memorizationin the teaching in Beijing. On the other hand, the finding concerningexaminationspoints to a common phenomenonin all threeplaces: the classroomteachingin all threeplaces was influencedto some extent by examinations.
AN EXPLANATIONOF THE DIFFERENCESFROMA CULTURALPERSPECTIVE
Two salient differencesin classroompracticeobservedin the threeplaces are thatin Beijing andHong Kong, teachingwas predominantlydone with the teacherexpounding mathematicalcontent in front of the whole class of students,and duringclassworktime, studentswere to practisethe skills just learned.In London, much more time was devoted to seat-work,and studentsusually learnedmathematicsindividuallythroughdoing the tasks in the textbook.Also, differentstudentswere usually workingon different tasks duringseat-worktime. , The different practice is of course related to the instructionalmodes used in the three places and the differentways mathematicsteaching is organisedin the three places. In Beijing and Hong Kong, with a class of 40 or more, it was difficult for the teacherto monitorthe progressof all the studentsif they were doing differenttasks and proceedingat different paces, and so the teacherhad to resortto whole-class instruction.And this way of organisationmay be due to limitationsin the financingof education. (Note thatthe expenditureon educationin Chinais much less thanthat in England.5)
These explanations,however, fail to explain the phenomenonof large class sizes in Hong Kong (and Japan and Taiwan), for although Hong Kong could affordto financeteachingin small classes, it seems thatit did not choose to do so. Also, althoughexpenditureon educationwas low in China, Chinese teachersenjoyed a much lighter teaching load than their counter-partsin Hong Kong and England.6Given the same amount of funding, smaller class size could have been achieved throughincreasing the workloadof the teachers(therewere otherresourceimplicationssuch as the availabilityof more teachingspace for smallerclasses, butthese did not seem to be importantfactors in influencingdecisions on class size). But this did not happenin China.
Moreplausibleexplanationscan be offeredat the culturalor ideological level. Given thatChineseteachersheld the morerigidview of mathematics being more a productthan a process, the most importantthing for them in mathematicsteaching was to have the mathematicscontentexpounded clearly.So therewas no need for smallerclasses, becausefor lectures,class size has little effect on the efficiency of delivery of content(this argument is similarto the argumentfor large groupsin lecturesat universitylevel). Moreover,if the assumptionof the desirabilityof differentstudentslearning differentthings was not valid in Beijing, the problemof large class size did not exist at all, because whereasthere were advocatesof attendingto individualdifferencesin London,the predominantbelief in Chinawas that studentsshould be taughtthe same thing in orderto be fair to them. In Hong Kong, the classroompracticewas similarto thatin Beijing, but therewere moreadvocatesof smallerclass size fromprogressiveeducators. So Hong Kong can be consideredas being in a state between Beijing and London, inclining more to the Beijing side, and this is consistentwith the culturalrelationshipbetween the three places discussed at the beginning of this paper. Anotherfindingthatdifferentiatedclassroomsin the threeplaces from each other was the amount of 'off-task' time and incidents of teacher maintainingdiscipline observed. 'Off-task' time included time when the teacherarrivedat the classroomafterthe start-of-lessonbell, whenteaching and learningstopped before the end-of-lesson bell, and when the teacher was engaged in administrationor maintainingdiscipline while the class was not workingon mathematics.These incidentshappenedmore often in Londonschools when comparedwith schools in Hong Kong, which in turn had more 'off-task' time and incidents of teachermaintainingdiscipline thanBeijing schools. It is clear thatthese differencesmighthave immense effects on students'learning.Of courseit was thequalityof theteachingand learningactivities which was of fundamentalsignificance,but if the actual quantityof time available for mathematicslearningdifferedsignificantly (the ratio of time spent in teaching and learningmathematicsin Beijing, Hong Kong and London, taking into account the lengths of the school year in the three places, was of the orderof 8: 7: 4), the studentswho received less instructions(the London studentsin this case) might be at a disadvantage. Again, the majorityof the 'off-task' time is relatedto the way classes were organizedin the three places. In Beijing, studentsof the same class stayed in a fixed classroom. There was always a five or ten minutebreak between lessons, andteacherswould usuallyarriveat the classroombefore the bell and startthe lesson as soon as the bell rang. In Hong Kong, for
junior secondary classes, students of the same class usually stayed in a fixed classroom as well (though there were cases where studentshad to move from one classroomto anotherbetweenlessons). But theremightnot be any "break"between lessons, and time had to be takenfor the teacher to move from one classroomto another. In London, mathematicslessons took place either in a special room or a room that was usually used by the particularteacher,and it was the studentswho were to move from one classroomto anotherand more time was needed for this moving. Also, since textbooks and all other learning materialswere provided in the maintainedschools, time was needed for distributingand collecting these materialsduringlessons. On top of these demands, there was time spent on other administrativeduties (such as taking roll call every lesson) and maintainingdiscipline by the teacher, and it was naturalthat this way of organisationwould give rise to more off-task' time. In the explanationabove, however,it is not clear thatif 'off-task'time was so undesirable,why were lessons in Londonorganizedin this way in the firstplace?Also, given thatall threecities understudyaremetropolitan areas, why were there greaterdiscipline problems in London?Again an explanationat the culturallevel is suggestedbelow. The notionof 'off-task'time in this studyis basedon the assumptionthat all lesson time shouldbe used for the teachingandlearningof mathematics. This is indeed an assumptionthatseemed to be held by teachersin Beijing andHong Kong. It was pointedout earlierin this paperthatfor the Chinese, learningis a seriousendeavour,and one is not supposedto waste any time for learning.The classroomis the place for learningandstudentsshouldnot be engaged in irrelevantactivitiesduringlessons. In Hong Kong, on top of these Chinese culturalvalues held by most teachers,the cult of efficiency thatis prevalentin the Hong Kong society mighthave also permeatedinto the classroom, and 'discipline' problemsthat got in the way of teaching were less toleratedby teachers.ForLondon,it seems to the researcherthata differentmentalityexisted in teachers.Study(andwork)was to be enjoyed, and lesson time was interpretedflexibly.Digressionsinto somethingother thanmathematicsteachingandlearningduringlessons were tolerated,and it was not expected thatthe whole durationbetweenthe start-of-lessonand end-of-lessonbells be used for mathematicsteachingandlearning.So in the finalanalysis,whatappearedto the researcherto be 'off-task'activitiesand disciplineproblemsmightnot be consideredproblemsat all for the London teacher.This was evident fromthe interviewswith teachersafterobserving theirlessons. When interviewingteachersin Londonafterobservingwhat the researcherconsideredlessons with disciplineproblems,very often the
interviewees would comment that they were satisfied with the discipline of the class which they had just taught. But the typical reactions from Hong Kong teachers would be to apologise to the researcherfor the bad behaviourof their students.(Therewas only one incidentof a lesson with discipline problemsobservedin Beijing.) Otherfindings for the classroom observationsare that Beijing lessons were highly structuredwhile there were more extemporaneousinstructional decisions made by the teachers in Hong Kong and London, and that there was more use of rigorousmathematicallanguagein the Beijing classrooms. The classroom practice of teachers in Beijing also reflected a less flexible view of mathematicsand mathematicseducation,a greater stress on conformity and a strongeremphasis on memorizationthan that of Hong Kong teachers,and in turnthe Londonteachers. The highly structuredlessons in Beijing and the meticulousexposition of mathematicsconcepts by Beijing teachersshow that they preparedfor their teaching thoroughly,and this was possible only because of theirrelatively light teaching load mentionedearlier.This thoroughpreparation for lessons may be reflectionof a feeling of incompetenceon the part of the Chinese teachers,or it may well be a traditionof what is expected of a teacher in China. Incidents of extemporaneousinstructionaldecisions made by Hong Kong and Londonteachersin theirclassroomsmight have causedthemto appearto be less well preparedin theirlessons, andthis was consistentwith the heavy workloadthey hadin teaching.Hong Kongteachers' stress on practisingskills may be a reflectionof the stronginfluenceof public examinationson classroomteaching.But these explanationsarenot sufficientin accountingfor the dominantpositionsof rigorousmathematical language,conformityand memorizationin Beijing teachers'teaching. Reasons for these characteristics,it is proposed below, lie in the views held by Beijing teachers on mathematicsand mathematicsteaching and learning. ForBeijing teachers,it seems thatmathematicswas perceivedas a fixed body of knowledgeto be learned,andif necessarymemorized,throughhard work.So the importantthingin teachingwas to have mathematicsconcepts expoundedthoroughlyand accurately,and this was best achievedthrough a carefully structuredlesson and throughusing a rigorousmathematical language.The Chinese nationalcharactersof conformityand compliance mentioned earlier also help to explain the uniformityof teaching styles observed in Beijing and the Chinese teachers' practiceof requiringtheir studentsto learnmathematicsin a standardway. The emphasis in Londonclassroomson learningmathematicsthrough doing it andthe use of non-technicallanguagein talkingaboutmathematics
also point to a differentunderlyingunderstandingof whatmathematicsand mathematicseducation are. For London teachers,mathematicswas more a process than a product,and it was somethingthat you learnedthrough doing it ratherthan a fixed body of knowledge impartedby the teacher to the students. Extemporaneousdecisions made in the classroom were not perceived as lack of preparation,but as respondingflexibly to the classroom situation.Rigorous mathematicallanguage was thoughtto be unnecessaryor even undesirablefor studentsat this level. For Hong Kong, it seems that the influence of both the Chinese and English views was present.But the most dominantview of mathematics and mathematicseducation seems to be that mathematicswas perceived as an importantschool subject that would help students move up the educationladder.So the importantthing was to help studentsto perform well in examinations,and the best way to preparestudentsfor this was throughgiving them ample chance of drill and practice. 7.
The classroom observationresults in this study portraythe mathematics classroom practices in Beijing, Hong Kong and London, and show that there are striking differences in classroom practices between the three places. Although this study is not meantto be a verificationof the cultural explanationof the differentpractices,the discussions above show that the culturaldifferences in attitudestowardsmathematicsand mathematics teaching and learningare potentiallypowerfulexplanatoryfactorsfor comparativecurriculumstudies. The culturalperspectiveis a factor that futurecomparativestudies in the mathematicscurriculumcannotaffordto ignore in interpretingthe results of theirfindings.
Eventsfor which Durations were Measured Whole-ClassActivities: Symbol Wo Wn Wb Wq Wa
Event Teacherreviews contentwhich the class has learnedpreviously. Teacherexplains new content. One or more studentswork on the board. Teacherasks questions,one or more studentsanswer. Teacherleads an activity,and studentsengage in essentiallythe same activity.
Groupactivities: Students (more than one third of the class) engage in discussions or other activities in groups.
IndividualActivities: Is Id Ir Ia
Individualstudentsdo the same class-work/seat-work. Individualstudentsdo differentclass-work/seat-work. Individualstudentsread(textbooks)in theirseats. Individualactivities otherthanthe above.
Teacher engages in administrativework while students are not working on mathematics. Teacherkeeps discipline or teachernot in the classroomwhile studentsare not workingon mathematics;confusion or other"off-task"behaviours.
Eventsfor which Occurrenceswere Noted A. EventsRelated to ClassroomPractice 1. Elementsof the teaching a. b. c. d.
Teacherannouncesthe title of the lesson or writesthe title on the board. Teachersummarizesthe contentcovered from time to time. Teachermakes extemporaneousdecisions in the teaching. Teacherrefersstudentsto the textbookin the explanationor follows the textbook closely in the teaching.
2. Teacherand studentsactivities a. b. c.
Teacherengages in administrativework (e.g. roll-call, announcements,distribution of materialsetc.). Teachermaintainsdiscipline (e.g. remindsstudentsto listen or to carryon with their work). A substantialnumberof students(morethan3) cause disruptionto the teaching, muck aboutor not engage in mathematicslearningactivities.
B. EventsRelated to Viewsof Mathematicsand MathematicsTeaching and Learning 1. Use of rigorouslanguage Teacheruses or requiresstudentsto use rigorousmathematicallanguage. 2. A flexible view of mathematics This includes: a. b. c.
teacheror studentsemploy alternativemethodsin solving the same problem, teacheror studentsuse a trialand errormethodin solving problems,and studentsengage in investigationsor explorations.
3. Conformityand a rigid view of mathematics These include: a. b. c. d.
teacherrequiresstudentsto follow the teacher'swordingexactly, teacherrequiresstudentsto follow fixed steps in solving problems, teacherrequiresstudentsto presentsolutionsin a fixed format,and teachercategorizesproblemsinto standardtypes.
4. Memorizationof mathematicalresultsand processes This includes: teacher requires students to memorize mathematicalresults or processes, or studentsrecite mathematicalresultsor processes. 5. Expectationon studentsto work hard Teacherurges studentsto study hard. 6. Examinations Teachermentions aboutexaminationsin the teaching.
MATHEMATICS CLASSROOMS APPENDIX II
Durations of VariousEvents in the Classroom(see the note on the next page) Beijing
Wo mean s.d. i.q.r.
6.68% 9.78% 7.40%
9.44% 11.05% 15.65%
7.84% 17.01% 9.20%
Wn mean s.d. i.q.r.
58.56% 12.49% 15.20%
53.06% 14.55% 19.70%
23.89% 15.46% 23.45%
0.00% 0.00% 0.00%
Wb mean s.d. i.q.r.
2.67% 4.89% 4.00%
4.88% 7.63% 8.55%
1.35% 4.98% 0.00%
0.00% 0.00% 0.00%
Wq mean s.d. i.q.r.
0.00% 0.00% 0.00%
0.11% 0.64% 0.00%
0.00% 0.00% 0.00%
0.00% 0.00% 0.00%
0.24% 0.91% 0.00%
0.00% 0.00% 0.00%
2.97% 12.95% 0.00%
1.79% 4.37% 0.00%
Wa mean s.d. i.q.r.
Sub-totalfor whole-class activities
G mean s.d. i.q.r.
322 Is mean
0.00% 0.00% 0.00%
0.00% 0.00% 0.00%
5.71% 14.95% 0.00%
77.84% 21.16% 22.40%
Ir mean s.d. i.q.r.
0.09% 0.50% 0.00%
0.00% 0.00% 0.00%
Ia mean s.d.
0.21% 0.85% 0.00%
0.00% 0.00% 0.00%
Xa mean s.d.
Xo mean s.d. i.q.r.
0.00% 0.00% 0.00%
7.46% 4.63% 7.30%
13.51% 9.75% 16.20%
12.06% 10.57% 22.70%
Sub-totalfor off-task activities
i.q.r. Id mean s.d.
London(W) standsfor Londonlessons conductedin a whole-classinstructionsetting, London(I) standsfor Londonlessons following individualizedprogrammes; s.d. standsfor standarddeviation;i.q.r.standsfor interquartilerange.
Occurrencesof EventsRelated to ClassroomPractice Events
Hong Kong (36)
London(I) (16) 0 0 0
1. b. c. d.
title summaries extem. decisions textbook
34 22 0 32
17 5 12 24
9 1 8 2
a. b. c.
administration discipline "off-task"
0 1 1
2 6 4
NOTES The alternativeof countingthe actualnumberof occurrencesratherthannoting whether the events occurredor not in a lesson had been considered.But since some behaviours were exhibited repeatedlyby some teachers (e.g. some Beijing teachers' use of rigorous mathematicallanguage), it would be exaggeratingthese occurrencesof the behavioursor events if the unit of analysis was numberof occurrencesratherthanthe numberof lessons (or teachers)exhibitingthose behavioursor events. 2
Because of the scarcity in resources,the Chinese adopta system of "key-point"schools whereby special resources are allocated to some schools which select the best students throughpublicexaminationsfor education.These schools are"thebest of thebest, favoured in facilities, teaching staff, and in funding"(Unger, 1982:19). The disparityin facilities between these key-point schools andotherschools is very considerable.This elitist system is "justifiedin terms of the need to identify and trainChina'smost talentedyouth in order to speed up the process of modernization,thus eventually benefitingthe entire society" (Hawkins, 1983:35). 3This was not counted as a group activity in this study unless the discussions were organizedby or carriedout at the instructionof the teacher. 4
One example of such administrativework was that for many schools in London, the teacher had to take roll-call every lesson, whereas in Beijing and Hong Kong roll-calls were usually done before the firstlesson began and so did not take up the teachingtime. 5The government expenditurein 1988 on education in the three places is as follows
(State EducationCommission (1991); Roberts(1989); Dennis (1991)):
tabcosep6pt China Hong Kong
GovernmentEducationExpenditure as a percentageof GNP or GDP
GovernmentEducationExpenditure as a percentageof public expenditure
GDP figuresin China are not available,and so GNP figuresareused in the table above. 6
In China, the average mathematicsteacher had to teach only two 45 minute lessons per day (Mondays to Saturdays,giving a total of 12 lessons per week, comparedwith the correspondingaverage figures of about 28 and 25 lessons in Hong Kong and London respectively), and the two lessons were usually "parallel"lessons on the same content so thatteachersonly had to preparefor one lesson per day.
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The Universityof Hong Kong, Departmentof CurriculumStudies, Hong Kong. Email: HRASLKS@HKUCC.HKU.HK