SN 1946 User Guide
Contents
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3. PROBABILITY CONCEPTS IN 11 16 YEAR OLD PUPILS Report of research sponsored by the Social Science Research Council 1978 1981 DR DR GREEN CAMET Centre for Advancement of Mathematical Education n Technology University of Technology Loughborough England Second Edition Published Aurust 1982 a b c a AIMS OF THE PROJECT The aims of this project which ran from November 1978 to October 1981 were a to survey the intuitions of chance and the concepts of probability possessed by English school pupils aged 11 to 16 years residing in the East Midlands region b to establish patterns of development of probability concepts and to relate these to other mathematics concepts supplementirg the work of the SSRC funded project Concepts in Secondary Mathematics and Science based at Chelsea College 1974 1979 Refs 1 2 c to investigate pupil responses to GCE O Level and CSE probability items with particular reference to sex differences TEST DEVELOPMENT A special Probability Concepts Test was devised over a two year period entailing six pilot versions before achieving the definitive form Appendix A DATA SOURCES Altogether about four thousand pupils were involved in the testing although ultimately only 2930 comprised the final Sample Pupils in East Midlands state comprehensive mixed schools aged 11 to 16 years i e in normal school years 1 to 5 were tested on a class basis with
4. 7 5 e Mansfield D Nevark H x NOTTINGHAMSHIRE e s DERBYSHIRE S 77 LINCOLN 1 SHIRE E 4 d Nottingham Pd y 1 Derby 2 P 2 Ne De Aa sey 2 m y N Pid 2 1 0 8 STAFFORD V Mai aert ver SHIRE N s 00 7 Loughborough Melton Mowbray PN 3 1 ae 4 N N RUTLAND H af Leicestet x l 2 4 E D E ERE 77 3 i z 2 LEICESTERSHIRE NN EN EE Corby e ame H H EN H M r 5 1 WARWICKSHIRE waar NORTHAMPTONSHIRE gt e Northampton Locat ons of the 44 schools n the Ma n Sample i 38 NOTE on GCE O Level and CSE Exam nat ons The GCE O Level Exam nat on s designed for the top 208 of pupils in English schools and s normally sat at 16 years of age The CSE Examination was intended for the 408 of pupils immediately below the top 20 However t is very often sat by pupils of much lower ability and sometimes by pupils of higher ability The age at which it is sat is 16 years i e after 5 years secondary education For both CSE and GCE O Level the probability questions set are of a formal computational nature involving sum and product calculations and the use of tree diagrams 39 Varna dd Aru dr CHANCE
5. 9 dM IG 2 47 59 Mu 6 reason Same 5 Ga 6 a reason G 9 yazan E G c reas bld 6 Mason G A Gle reason 4 or 3 fi 2 i0 3 iv 2 8 0 MARKING SCHEME Sulake Pe ee pe P6 6 Pe 6 Pe PB eG 66 eg ve Ve V P Sept 1932 Qu lo 4 r n cC or Band C 2 or and 6 Ib B E 7 2 17 reason Ie 8 40 1466 2063 Clare b 216 A 2160 A 21 e 2144 22 1 or 2 23 8 3 by aad extra tides or if i above missed CL ee 6 ye ye reduced PB Pa PB PG pa PB Pe 68 pe 6 6 PB 48 Ona hm Sept 482 Mak Sublet 24 5 ea 25 ea 26060 ABC ACB BAC BCA CAB CEA CB C 3 24 c a 120 ce VB tal IS Verts ui CB h t 5 Cu sal PB 30 50 MARKING _ SCHEME MSZ 49 Codes QN Code 57 gt 2 Be c 2 4 7 5 3 Ge Cb uGe 3 Cd 2 Ge Jai 4 Tau 2 Ta tit 3 4 7a iv 2 7bi 1 2 3 4 761 135 Tbiv 7 bv 3 3 q 5 lo 1 3 E 35 12 2 13 1 4 l4 Is
6. ye e ed 13 G m AH 55 RoTH Ala GOTH BSu 21 50 0 22 2 iss di nary 3 5 25 26 oc Reas Same 27 29 7 5 2 8 ES K r Es ped 22 wack hore Ole ama Qo y CC frato 7 B K A taa 6 J moe 00 ao goz MC M n n 3 gt 7 uti bago 3 sex chance gt gt more 5 compared As Gg Bot B Bota Lamela BTH bane 5 vu 7 CH Loy W 2 U K 6 dede 5 sama bur cad LUCK mare now ratio 2 UAM iO y pl proportion Ke 207 61 7 ener part MCR 0 Ae 5 y U OMIT RS ele E ev 5e tannet happen d isst ET E its ra paralele E antr 8 Chance KARI R m ble e 7 7 ln E bu nek Aard SC SE er vr m veldes Can com be eff af o sabel udik aet sd p 7 5 by mart ee SI oey taag pot Hap oer Core Omar 2 rege certain SES 5 62 E
7. amp PROBAETLITY CONCEPTS PROJECT Details of Classes we wish to Test Ability Probability Mathematics Teacher s Assessment Teacher 3 Teaching Style Secondary School Level years 1 to 5 inclusive age 11 16 years We are interested n testing right across the ability range Wherever possible we would like to test mixed ability groups We do not mind whether or not pupils have been formally taught probability Our test makes no reference to the term probability We do not mind if the pupils are in mathematics sets tutor groups streams general studies groups etc It would be very informative for us if the school could provide a broad assessment of each pupil s mathematical ability Bow this can Le best done vill be discussed when we visit Wu are interested to see if there is any correlation between teaching style experienced including used and understanding of probability concepts Me hope to obtain that information during our visits D R Green 1980 2 CHANCE amp PROBABILITY CONCEPTS PROJECT Teaching Scyle Questionnaira School Date Class Set Stream in relation to whole year group Textbook s used dn When waa work on Probability last douer What topics were covered When was work on Fractions last done e What topico were covered
8. What average of lesson times is dovoted to a Presentation of new material including worked examples b Pupils solving prolema working in groups c Pupils solving problems individually d Going over homework going over difficulties etc etc What average of lesson times is devoted to practical work using apparatus or act vities e g using compasses rulers counters scales dice geoboards taking censuses CHANCE amp PROBABILITY CONCEPTS PROJECT 1 Deta ls of the Tests Probability Concepts Test Duration minutes This comprises 26 questions which seek to establish the level of understanding uf the pupil on random events permutations combinations and arrangements No knowledge of formal probability is assumed Zach pupil will be given the 16 page Test Booklet For all pupils in years 1 2 3 i e aged 114 12 134 the test will be read out This is to minimise reading difficulties to ensure appropriate emphasis of the important ideas in the questions and to ensure that each question receives the same attention from each pupil Our experience with pilot versions of the test indicate that 60 minutes is adequate for all groups We shall not normally read out the test questions to pupils in years 4 5 i e aged 14 15 unless the school recommends doing so Intelligence Test Duration Introduction 18 minutes Test 42 minutes The chosen test is called AH2 It
9. Bag F 4 black and 4 white F O ee 0 oo Which bag gives a better chance A Bag E L i B Bag F C Same chance D Don t know Why 2 CHE os Of picking a black counter Two other bags have black and white counters Bag G 12 black and 4 white Bag H 20 black and 10 white Which bag gives a better chance A Same chance of picking a black counter B Bag G E1 C Bag H E D Don t know L Why v S Two other bags have black and white counters Bag J 3 black and 1 white Bag K 6 black and 2 white 1 Which bag gives a better chance A Same EG Ee B Bag E C Bag K 1 5 Don t know Of picking a black counter A5 7 a Here are five phrases Cannot happen Does not happen very often Happens quite often Happens nearly every time Always happens For this question you must put one number in each of the four boxes i Which 11 Which 111 Which below You can use the same number more than once if you wish of these phrases means the same of these phrases means the same ot these phrases means the same iv which of these phrases means the same 7 b Give a word or phrase which means the 2 Impossible 5 2 11 Possible 111 Even chance 1v Little chance v Very probable as Very l kely as Unlikely as Likely 0000 as Not very l kely 0 same as the follov ng 21 1
10. 01011001100101011011010001110001101101010110010001 01010011100110101100101100101100100101110110011011 01010010110010101100010011010110011101110101100011 SUSAN 10011101111010011100100111001000111011111101010101 11100000010001010010000010001100010100000000011001 00000001111100001101010010010011111101001100011000 Now one girl did it properly by tossing the coin The other girl cheated and just made it up a Which girl cheated ANSWER b How can you tell ANSWER 21 a Suppose a lot of marbles are dropped down the set gt A12 below n m 12345678 Tick the sentence which best describes where you expect the marblrs to go A Each channel will get about the same number of 8 1 and 8 w111 98 most marbles L C 3 4 5 and 6 will get the most marbles D 1 3 5 and 7 will get the most marbles E None of these b Nov do the same for this set of channels i 6 1 23 4 A Each channel will get about the same number of marhl 8 1 and 2 will get the most bx C 3 and 4 will get the most 5 1 and 4 will get the most 3 a E None of these Of channels drawn 28 tc d Nov do the same for this set of channels A B C D E Now do the same for this set of channels A B C D E 1 2 7 A13 3 Each channel will get about the same number of marbles 2 will get the mos
11. 1 7 5 4 Clare Susan Cannot tell Omit 1 38 53 1 7 2 34 56 2 7 3 30 59 4 7 4 40 46 3 9 5 35 49 5 9 All 35 53 3 8 1 2 3 4 5 6 8 Correct Pattern Inequal1ty Equal1ty cols 1 7 Regular1ty of 1 of 0 1 Lengths Other 6 c S c 5 17 13 14 16 12 14 Ar B 5 E Qn 21 b Year Ar B 6 0 1 48 21 12 8 11 1 34 17 23 17 2 52 22 11 7 7 2 37 16 17 22 3 57 18 10 6 8 3 44 13 13 20 4 57 20 8 3 10 4 46 11 12 17 5 65 14 5 3 11 5 53 11 11 13 All 55 19 10 6 9 All 42 14 16 18 Year A B 5 E Qn 21 d Year A B D S gt 61 ig 4 1 29 36 10 18 3 is i 69 11 3 2 24 37 11 21 7 isi 68 12 5 3 22 41 7 24 67 14 7 4 18 42 5 29 5 2 71 10 6 5 17 50 5 22 12 2 67 13 5 38 an o 13 14 14 11 12 13 1 1 t woo m 2 o Tun uy nun BS Qn 22 Year 1 and 2 ti or 2 5 6 7 8 Same Other Omit 1 5 7 22 57 6 3 2 4 5 20 62 6 2 3 7 3 19 62 7 3 4 9 7 16 59 7 3 5 13 4 15 59 5 3 All 7 5 19 60 6 3 Qn 23 Year A Bt D E Qn 24 Year A Dr D 9 15 8 9 58 1 18 7 8 66 3 8 16 9 6 60 2 16 9 5 69 4 5 15 8 6 65 3 25 7 4 63 5 4 18 8 4 65 4 33 8 3 54 3 20 7 4 66 5 41 6 3 49 ALL 6 17 8 6 62 All 25 8 5 61 Qn 25 Year A B c 1 13 63 24 2 13 68 17 3 13 73 13 4 18 68 12 E 5 25 63 10 All 16 67 16 26 8 correct and omits only a b 6 9 Year 6 Omit 24 Omit 120 Om t 1 64 4 5 17 1 21 2 68 3 9 16 2 21 3 81 1 15 8 2 13 4 80 2 19 9 9 15 5 87 4 25 9 9 13 All 75 3 14 12 4 17 APPENDIX
12. 29 14 22 31 3 1 37 10 28 21 4 2 40 7 26 25 3 2 43 6 36 10 4 3 40 6 25 25 2 3 42 5 41 10 3 4 50 8 19 20 3 4 46 6 38 7 3 5 48 11 21 17 2 5 44 8 39 7 2 All 41 9 23 24 3 All 42 7 36 11 3 QN15 QN16 correct only Year Correct Ambiguous Other Omit Year A B C D E F 1 41 1 38 20 1 35 68 62 75 75 91 2 55 2 26 18 2 34 73 69 82 79 92 3 2 2 23 1 3 26 81 72 87 86 95 4 74 2 15 10 4 26 81 87 87 89 97 5 73 1 22 5 5 25 87 78 94 93 99 All 60 2 25 14 All 30 77 71 84 84 94 t QN17 Year A B EN D 1 17 29 49 4 2 22 26 49 3 3 20 26 53 1 4 26 22 50 1 5 21 18 59 2 All 21 25 52 3 Contiguity Idea Proba R son Year Area Counting Equal Close Spread Ratio b lity Other Omit Concept Concept Better Better Concept Concept MC MM M i tL 1 1 35 1 21 14 0 0 17 10 2 3 34 2 21 18 0 1 16 7 3 3 35 3 21 17 1 2 12 6 4 4 31 3 19 22 1 1 8 11 5 5 38 3 14 19 1 4 7 8 ALL 3 35 2 20 18 1 1 12 8 Qn 18 An 19 Qn 20 a Year Qn 20 b Qn 21 a Year Qn 211 c B4 Year A B 5 E Qn 19 Year A B 1 7 40 12 38 3 1 32 43 17 7 2 6 45 9 38 3 2 27 49 18 6 3 3 61 5 30 o 3 19 61 16 4 4 4 65 3 25 2 4 18 63 16 3 5 2 70 3 22 2 5 14 69 14 3 All 5 55 7 31 2 All 23 56 16 5 4 Reason Year Area Counting Equal Ratio Contiguity Position Other Om t Concept Concept Concept or speed 1 41 27 7 1 7 1 10 6 2 51 24 5 o 3 2 8 5 3 62 15 4 1 5 2 6 6 4 67 14 4 o 4 1 5 4 5 69 13 2 2 4 o 3 6 ALL 57 19 4 1 5
13. 3 4 5 1 20 14 61 2 1 3 16 23 40 16 2 21 16 58 3 2 4 6 20 44 22 3 35 16 46 2 3 3 7 13 57 18 4 42 13 44 1 4 1 6 14 57 21 5 56 9 33 1 5 o 4 11 68 15 All 33 14 50 2 All 3 8 17 52 19 Qn 7 a 14 n 7 a Qn 7 a av Year 1 2 Year 3 4 5 Year 1 2 1 38 50 1 38 33 20 1 27 55 2 31 60 2 45 28 18 2 29 57 3 30 63 3 53 27 15 3 25 66 4 31 65 4 53 31 13 4 25 70 5 25 72 5 60 25 11 5 22 73 All 32 61 All 49 29 16 All 26 63 Qn 7 b correct only ES LA NEM DAP Yer 4 di Gi av wv Qn 8 Year A oct D E 1 84 73 18 65 23 1 4 4 11 8 73 2 84 76 25 69 27 2 4 4 11 6 74 3 89 80 34 78 37 3 1 4 16 6 73 4 89 85 38 83 43 4 2 4 12 5 77 5 91 91 44 84 51 5 o 3 21 6 69 All 87 80 30 75 35 All 3 4 14 7 73 Correct 9 10 Qn gn v th Year 1 2 3 4 5 6 Other Omit Year Correct Repeats Incomplete Omit 1 19 6 5 3 46 8 9 3 1 42 7 42 8 2 17 5 2 2 51 11 8 2 2 51 7 34 9 3 13 4 3 1 62 9 2 2 3 64 5 25 6 4 9 2 1 1 68 11 6 2 4 69 3 25 3 5 9 2 2 1 70 11 4 1 5 81 2 13 3 All 14 4 3 2 58 10 7 2 All 59 5 30 6 94 0811 0912 Year B Band C equal Year A Br E A and B equal A A 1 11 12 26 23 27 1 14 31 15 19 21 2 13 12 26 20 29 2 14 30 13 19 23 3 13 14 24 25 24 3 15 29 13 25 18 4 13 16 16 21 33 4 18 24 14 19 25 5 12 15 18 21 34 5 14 23 12 23 26 All 12 14 23 22 29 All 15 28 14 21 22 QN13 QN14 Year Correct Ambiguous Certain Other Omit Year Correct Ambiguous Impossible Other Om 1
14. 6298 ar In the diagrams Yellow Red LOY SE UN XD Which spinner gives you a better chance of landina on a 2 when ot spins or do they give the same chance A Yellow 1s better tor getting A 2 B Red is better for getting a 2 L C Both spinners q ve the same chance D Don t know crc EES 18 4 red marbles 4 blue marbles and 2 green marbles are put into a baq which is then shaken Three matbles are picked out 2 red and 1 blue Then one more marble is picked out Which colour te if most Liki ly to be A Red has the best chance B Blue has the best chance C Green has the best chance D All colours have the same chance 00000 Don t know 10 All 19 a Two discs one orange and one brown are marked with numbers VN 22 Brown Orange Each disc has a pointer which sp ns round If you want to get a 1 1s one of the discs better than the other or do they both give the same chance A Brown is better for getting a 1 B Orange s better for gett ng a iL C Both discs give the same chance one can say 19 5 Why did you choose this answer 20 A teacher asked Clare and Susan each rd toss a coin a large number of times and to record every time whether the coin landed Heads or Tails For each Heads a 1 is recorded and for each Tails an O is recorded Here are the two sets of results CLARE
15. GIRL gt AS CAMET AGE O EES DATE OF BIRTH Loughborough University of Technology MATHS SET M YEAR DR Green 1980 NAME 204 soo 27 1 A small round counter s red on one s de and green on the other side It 35 held with the red face up and tossed high in the air It spins and then lands Which side is more likely to be face up or is there no difference Tick the correct answer A The red side is more likely 8 The green side is more uxey There 15 no difference D Don t know 2 A mathematics class has 13 boys and 16 girls n it Each pupil s name is written on a slip of paper All the slips are put in a hat The teacher picks out one slip without looking Tick the correct sentence A The name s more likely to be a boy than a girl Es B The name s more likely to be a girl than a boy ES It is just as likely to be a girl as a boy D Don t know 17 3 Here are pictures of two discs which have pointers which are spun and point to a number With which disc is it easier to get a 3 Tick the correct answer RED BLUE A It is easier to get 3 on the Red disc B It is easier to get 3 on the Blue disc C The two discs give the same chance of getting a 3 C D Don t know Why did you chose this answer 4 When an ordinary 6 sided dice s thrown which
16. found The relevant breakdowns are presented in Tables 8 and 9 The general trend is that whereas the more able pupils gain about 1 level during the five year period the least able starting 1 level behind falleven further behind during the same period only gaining about 4 a level See also Fig 1 The average pupil enters secondary School aged 11 just at Level 1 and leaves aged 16 at Level 2 AH2 Grade Year A B c D E M MM MM 0 62 0 80 1 08 52 1 1 84 1 0 67 0 86 1 30 1 77 2 15 2 0 67 1 29 65 1 2 12 2 54 3 0 77 1 25 1 89 2 27 2 67 4 1 10 1 50 10 2 2 44 2 83 5 0 74 1 10 1 55 1 98 2 36 1 A ee 4 Table 8 Mean Probab l ty Concept Levels by Year and by General Intelligence AH2 AH2 Intelligence Band Year A B C D E AAA 1 2 1 0 1 2 2 1 3 2 3 1 2 1 4 3 2 1 5 3 2 1 Table 9 Most typical Probab l ty Concept Level for each Year and Intelligence band Mean Probab l ty Concept 2 5 Level 1 5 AH2 Grade AH2 Grade AH2 Grade A top 104 B next 20 C middle 40 0 next 20 E bottom 10 Fig 1 Mean Probab l ty Concept Level rofiles b School Year and school Year eneral reason n ab l ty CROSS REFERENCE TO CSMS Some pupils in the Sample were subsequently further tested with a CSMS Fractions Test N 286 and or a
17. is an appropriate model which d ctates his choice of response Is colour a relevant factor Is past history Question 5 is similar A careful consideration of pupils responses to these two questions 1s recommended Are the suggested correct responses in fact correct Are there other defensible answers School mathematics very much emphasises mathematical manipulation and gives very little weight to either model building or interpretation Is this a proper emphasis 13 c d e f g h i 3 2014 Individually interviewing pupils across the whole age and ability range has confirmed that some pupils are quick to answer questions intuitively with no obvious intervention of mathematical processes Qns 1 3 5 6 particularly However other pupils lack this ability and it is clearly unwise for the teacher to assume that all pupils can see what to the teacher may appear obvious even if some in the class effortlessly provide the correct answers Only an extensive programme of class based activities 35 likely to provide the weight of evidence and experience to eliminate the fallacious thinking exhibited by pupils and adults alike Qns 1 4 5 17 20 24 25 particularly The more standard aspects of probability clearly require an understanding of fractions and ratio Ratio in particular is an essential prerequisite of dealing with the more complex situations Qns 2 6 The provision of diagrams seems
18. is published by the National Foundation for Educational Research NFER and is based on tha work of Dr A W Heim and others at the Cambridge Psychology Laboratory The test is in three parts Part 1 Verbal 15 minutes Part 2 Numerical 15 minutes Part 3 Perceptual 12 minutes Norms are available for secondary school pupils for each separate section and overall He ara using this test for two main reasons a to ensure the noraality of our sample b to investigate the relationship between probability conceptual level and Verbal Numerical and Perceptual ability Our original intention was to use a similar test 4 but AH2 has just been made available to us and is better for our purposes outweighing tha inconvenience of its being more lengthy to administer than AH4 2 Results of the Tests If the schocl wishes we will forvard to the school the test results of individual pupils although we recommend that they not be divulged to the pupils themselves D R Green 1980 5 R E d x RADE PROB TEST n v Je hrraserasio 023496 99 9 1 023 4967959 0934967999 053448789 6 12345678910 ER 123456 78910 12345678910 12345678910 67 ILL 2349878999 _ 6 asi 7 11 0534557995 m w d ja ln je o
19. more thar one ticked T lege 7 E A gt ES maam 0 1 0 7 Zb oO omt 57 77 GE aren 4 7 ilan Chisco 5 x 5 2 peaikt ig 5 carte or spard eg harder side m 2 naan 7 en ga 7577 ali q Urq justis te 52 MCZ ON AN WN REM A 20 0451 e G ll x s x si ME E a 8 m _ 0 mere AA 5 99 5 EN met Black CG A ond cane Wda CG lar Bade ul ms be Black v B ad sema Lite bett Sere Whit CL beth co EMT E feni Metin cetro A 2 Y az 8 0 757 aach m lees cata limt deme Prem i 56 ON o 6 1 2 13 14 E 1G 17 1 19 21 MC3 mev tha ut Acte caat Se Were Counting re A tse en beth daade Landen z slk 7775777 k M pg mee B t 20 posikonal argumant silt ratio 7 m bru de _ n 3 a 2 257 Ci sacas of as Su PWNS mbo fraction dem more Ult A D me Mb e D ad a Lisz 7 7 w both en in D 2 gt 3 Sien 19 m OIE 16500 mat Blake thar Mut
20. reading out would be so controlled as to give the same amount of time to a question for all pupils v The test would be read out to any 4th and 5th year classes for which this was deemed advisable by the class teacher Alternatively adequate time would be provided 55 minutes minimum for silent working PROBABILITY CONCEPTS TEST RESULTS Pupils responses to the Test Appendix A have been analysed by School Year and are presented in Appendix B A summary 1 included in Section 10 As a first step towards further analysis of the responses to the Concepts Test each question was allocated to one of three sub test categories as follows Verbal VB 7 13 14 15 16 Combinatoric CB 9 10 26 Probabilistic PB 1 2 3 4 5 6 8 11 12 17 to 25 Each question or part attracted a mark of if correct and O if incorrect yielding three sub scores maxima VB 15 CB 6 PB 29 and a total Test Score maximum 50 For each pupil these four statistics were computed and broken down by School Year Sex AH2 Grade and Mathematical Ability MA The test Score breakdowns are shown in Table 4 TEST SCORE VB SCORE CB SCORE PB SCORE Bov Girl no Girl Table 4 Mean Test Scores broken down by School Year and Sex 4 An indication of the relative importance to Test Score of age intelligence and sex 1s provided by the Analysis of Variance ANOVA results presented n Table 5 This shows that all are significant factors However the perc
21. 0 In an experiment 12 coins are AR all tossed up in the air together and land on a table If the experiment is repeated a lot of times vhich one of the folloving results vill happen most often A 2 heads B 5 heads C 6 heads D 7 heads and 10 tails and 7 tails E and 6 tails and 5 tails L E All have the same geg Mark and Steven play a dice game Mark wins 1 penny if the dice comes up 2 or 3 or 4 or 5 or 6 If it comes up 1 Steven wins some money How much should Steven win when he throws a 1 if the game is to be fair h z ANSVER A 2p coin and a 10p coin are tossed together One possible result Heads on the 2p coin and Tails on the 10p coin has already been put in the table H for heads and T for tails Write in all the other possible results 11 The flat roof of a garden shed has 16 square sections It begins to snow At first just a few snowflakes fall then after a while more have landed Below are Lhree sets of two pictures Each set shows the pattern of snowflakes building up first 4 flakes then 16 flakes SET A SET B SET C QULSIION Which of thesc sels best shows the kind of pattern you would expect to see as the snowflakes land Set B Set C Sets 8 and C Each kina of pattern 1s as likely AE 12 Below are three sets of three pictures Each set shows the pattern of snowflakes
22. 1 Probability Concepts Test about 1 nour Appendix A 11 NFER AH2 Test of General Reasoning Ability about 1 hour Refs 3 4 The mathematics teachers of the sample pupils were asked to supply details of each pupil s mathematical ability M A on a ten point scale and also to indicate the texts used and style of teaching adopted CSE Mathematics Mode 1 scripts were examined with the permission of the East Midland Regional Examinations Board 1978 N 1024 1979 N 959 1980 N 1068 Responses to the probab l ty tems vere coded and computer analysed So far the ma n investigation has concerned sex d fferences and much vork rema ns to be done n the future GCE O Level Syllabus C Mathemat cs multiple choice responses were supplied n punch card form by the University of London School Examinations Department as follows 1977 N 654 1978 N 779 1980 N 1089 Analys sof the probability items responses vas undertaken with particular reference to sex differences SAMPLE DETAILS From the original four thousand subjects those absent for one test were eliminated and from the remainder a stratified sample was drawn to reflect the proportions of the various intellectual levels in the English school population as a whole This was based on the AH2 grade distribution which is purported to be Refs 3 4 as follows A B 3 5 E Top 10 Next 20 Middle 40 Neyt 20 Bottom 10 The final Sample N 2930 was as indicated in T
23. 9 a fo 19 b O Geet 5 2 5 0 P bex A ara sir ssl are compl OLS az E saz bala 50505 specified 50 90 25 chose Aat ekigas concent 25 spud Lale nm M 75 t pido E E 1 OMIT perm r di Usame chance 2 0 o O Z 5 2 4 g O 2 3 4 s tas M 7 8 7 21 rad pot o 2 2 MCG OMIT Clare Sucan va BEE b ti el WES aia OMIT E SA DRA too Cor Clave ski Chores Too 50 3 Tee fo Sunan 0 522 Lors 50 of 25 50 da ua Clare m ahat runs Long nand bbe PE ima notla T no es sira faal M 70 22 0 MCI7 OMIT oa 2 27 others har dek 2 2 3 5 6 7 ec 7 ax 0 008 ear loa 8 70 ag he Ha DU de De 25 wpwpro OMfT A 6 c i dapande on nikiel 3 77 ura 71 zo a 8 OY p OOO OM T como re pa sx distinct uv he is p ABC once Cere Ant Pr
24. AA ene EN ETT CARD A 55 1 3 FEE 2 S4 OQ IS gt 8 ERROR E A 1 ST Bel N 7 SS M Eni E 8010 A ls _ 66 __ _ 016 a EEE o m _6 L ie sare 9 un m o 62 Qa lble 2 1 S 6 Tan 1 MM Qu 6 bs Q GF 66 Qu anar MONS A m 67 m ze Qu aces 8528 N 69 Qu 19 a 70 Qu 71 Qu 20 a 72 A 20 6 13 uz 7 B 5 EN 15 3 Qu 21 c EE S 2 An a 0 A Qut 8 78 23 71 An 24 xin SD 225 46 O78 loz 7 7 75 11 20 T ge d NAN _ d amp N Pea dis Teast Date di goed ds 219281 M 2 for E L te D WS 22 27 22 30 Pk d l Motin a De Bes EN 5 AID 5 Ax 2G la AG 26 9 e Q 26 9 29 68 Shat or lorg AM el aa perd 78 79 Auz LNNN Total Score AHZ dealt da Vell Nitel N Secre NN 40 _ NNN dd M an Meto Me NN E LO 0 N _ Seems a ANN 20000 DENE o E x
25. CSMS Ratio amp Prcportion Test N 169 Thereby an approximate match between CSMS Test Levels and the Probability Concepts Levels could be effected Table 10 For this purpose a Probability Concepts Level 4 vas used which comprised the combinatoric items 26c 26d correct answers to both parts being required See also Figs 2 to 5 which relate to CSMS Levels in Hart 1981 Chapter 13 Associated CSMS Level for given Test Fractions 1 2 Fractions 3 4 Ratio amp Proportion Probability Level Corresponding CSMS Stage Table 10 Match between Concept Levels n Probability Concepts Test and in CSMS Tests CROSS REFERENCE TO PIAGETIAN STAGES By making a detailed study of pupils responses to the comparison of odds items Question 6 and relating this to Piaget amp Inhelder s own work Ref 7 1t became possible to indicate the ages at which Piagetian Stages are normally achieved The results for three levels of general ability are shown in Table 11 It must be concluded that most English school children do not achieve the level of formal operations which is Piaget s Stage 111 or the Probability Concept Level 3 a result in line with findings on Science concepts reported by Shayer Kuchemann and Wylam Ref 8 Corresponding 5 5 Question 6 Sample Item Top 30 Middle 40 Bottom 30 IB a lt 11 lt 11 11 IIA b 11 13 16 IIB c 12 14 III d e 14 Table 11 Ages at which Piag
26. HOICE LONDON This line of research followed up the earlier work of Wood and Brown Ref 9 to which the interested reader is referred 1 Probability items are not generally among the harder items 2 Boys performbetter than girls on the mathematics paper 3 Boys perform better than girls on probability items 4 Probability items do not have different ranking on facility for boys and girls 5 The percentage difference in facility for probability items between boys and girls is greater than that for the whole paper 6 Boys below the median level perform better on probability items than do girls of comparable mathematical ability 7 Boys above the median level do not perform better on probability items than do girls of comparable mathematical ability SUMMARY OF FINDINGS RELATING TO CSE MATHEMATICS MODE 1 EMREB 1 Boys achieve superior grades to girls 2 Boys achieve superior scores to girls on the Syllabus 1 options B and C examination 3 There is some evidence that boys achieve superior marks to girls on the Syllabus 1 probability questions 4 Probability items do not exaggerate differences in mathematical ability between the sexes SUMMARY OF FINDINGS RELATING TO THE PROBABILITY CONCEPTS TEST 1 Nearly all test items show an increase in facility with age Marked exceptions are Qns 11 12 24 2 Nearly all test items show an increase in facility with intellectual ability 3 Often Years 1 and 2 display quite simi
27. Iba 0 IG c 0 CORRESPOND To CoRKECT ON 64 IG e 0 7 7 reason 8 4 19 Mason 20a 20b 215 21b 2 21 4 22 23 24 25 26a 26 b 26 c 26 d ANSWERS CODE WO 9 N wW NN 24 20 50 Moy Cobes WHICH ERESPOND To ANSWERS fre REASON RE N Gm hee Le En d be 7 2 10 22 24 27 29 29 ge G 1 2 1S 26 30633 1 51 GUIDANCE ON MARKING SCHEME MSS Valid amam weed elisa ora Suze Space angle or number Se Sato or waa fraction rbe lod axes disa pee of potir oc sped say ore Ga Sana Sica b l a 5 we say hecanae vo buf eh tum D G 4 Maan Valid Qmawer gt uml discus talis ar prpection prin pegarla er actual rum ath o des lis Sa G6 becouse tas Han Litt G because s mere Black Compared 6 The odd we gelir fer DAL 4 amantes umdd discues difference Un Mmumbero of Black mly o Black White differences Luck ek NOTE inlarpretatim n uau nado u A Aa ade box nn He 20280 52 MS GUIDANCE MSG G le Mason VoL Ao mitad discus O prpertos ik Som fc prrhape a Es Thay hare B Buck bitt have Ha sane Blac
28. a rego 6 Un oa be 7 net buz 7 word Life etter Selina for Gua olan he ds maha __ g othe Ub semi ev Q 7 lag Ll TE 14 Ceoc DIGIT Second cade DIG um OMIT SS comal eaa S x e 42 ao bodur imposa ble 1 ut vk giae al seta RA Bi v 0 JO b Be 1 66 LN MCI3 FiesT Cebe 7 Cerrtctinaao of amar O N Seq uk 00 OMT cork wentar tan Mee 7 mal c w chance z e moat acordado pe garbtics Te dee SES 5 evet 2 Core alafa E n E 77 do SE otter bat ante an Zo SELOND Cede DIGIT Catt of auawe 7 coins RA Bee Bond em a ar ger Acadeds 0 basti R esti 7 sus anes Wedde 7 rents Odur 5 gb agile nates 67 HCH 15 THIRD CODE 77 Rank 75 CUNT 00 s l madiun Sha ovan acudent p lt re can Crab winning s 4 5 25 5 77 E COT ee 570 N NO T4 68 box net ded 16 alip de En 5 Z ud 25 3 4 5 G 7 g 18 ao Lef 1
29. ables 1 2 3 and Appendix C AH2 Grade School Year SR A B e D E Overall 1 64 128 256 128 64 640 2 67 134 268 134 67 670 3 67 134 268 134 67 670 4 54 108 216 108 54 540 5 41 82 164 82 41 TOTAL 293 586 1172 586 293 2930 Table 1 Sample numbers by School Year and by AH2 Grade School Year Derbys Leics Northants Notts Staffs TOTAL 1 468 73 99 m 640 2 86 287 94 178 25 670 3 23 474 45 112 16 670 4 120 283 43 94 540 5 2 258 61 68 410 Table 2 Sample numbers by School Year and by County Age Range for Number of Number of School years Schools n Sample Pupils in Sample 9 13 7 295 10 14 2 146 11 14 12 973 11 16 4 293 11 18 10 729 13 18 2 95 14 18 7 399 34 2930 Table 3 Sample numbers by School Type iis All the part c pat ng schools were selected from volounteers obtained by writing to allrelevant schools within 50 miles of Loughborough PROBABILITY CONCEPTS TESTING PROCEDURE In order to minimise possible reading difficulties and to ensure that pupils attended properly to each of the 26 questions of the test and to provide comparable conditions to all classes tested it was decided that i The test would be administered by members of the project team 11 All testing would be preceded by a prepared introduction explaining its nature and purpose 111 The test would be read out to all 1st 2nd and 3rd year pupils reading time 48 minutes iv The
30. amp SIMMONDS V 1978 GUILFORD J P 1965 NIE N H et alia 1975 PIAGET J amp INHELDER B 1975 SHAYER M KUCHEMANN D E amp WYLAM H 1976 WOOD R amp BROWN M 1976 Project On Statistical Education 1980 FISCHBEIN E 1975 GREEN D R 1982 o dary School Children s Understanding nematics CSMS Cl sea College University of London Chiliren s Understanding of Mathematics 11 16 John Murray AH2 and AH3 Parallel Tests of Reasoning Brit J Psvchol65 1 11 AH3 AH3 Manual NFER 2nd Edition Fundamental Statistics in Psychology and Education McGraw Hill Statistical Package for the Social Sciences 2nd Edition McGraw Hill The Origin of the Idea of Chance n Children Routledge amp Kegan Paul The Distribution of Piagetian Stages of Thinking in British Middle and Secondary School Children Brit J Educ Psychol 46 164 173 Mastery of Simple Probability ideas amon GCE Ordinary Level Mathematics Candidates Int J Math Educ Sc Technol 7 3 297 306 Teaching Statistics 11 16 Statistics n Your World Foulsham The Intuitive Sources of Probab l st c Thinking n Children D Beidel Probab l ty Concepts n School Pup ls aged 11 16 years Unpublished PhD Thesis Loughborough University of Technology Library APPFNDIX_A SCHOOL ii Le 5 roo CHANCE AND PROBABILITY CONCEPTS PROJECT TODAY S DATE 5 d BOY
31. at we are surrounded by random occurrences every day of our lives Is it reasonable to treat so lightly such a pervasive property of our world It might be thought that with the emphasis given to symmetry in school mathematics pupils would have little difficulty in handling problems where symmetry was a key factor That this is not nessarily so is exhibited by responses to Qns 6c 16 17 21 This may reflect the gap between a theoretical study and its application 12 14 k 1 m x ASL Although Fischbein Ref 11 has shown that the multiplication law can be well understood by a careful teaching programme utilising tree diagrama our evidence is more in accord with most research which indicates that the multiplicative principle is poorly understood pupils having little in the way of basic intuition on which it can be based Response to Qn 22 was very poor Even pupils who had recently studied tree diagrams said 1n interview that tree diagram work did not help them i e was not relevant Drawing inferences from sampling experiments was another very weak area as judged by responses to Qns 23 25 Pupils tended to rely on the abstract idea of equality and not relate it to the physical situation A lack of experience in this area may be a factor clearly pupils need to be given such experience and the interest shown in the media for opinion polls is clearly am excellent basis on which to build The stability of frequencies
32. building up first 4 flakes then 16 flakes then 64 flakes SET A SET B SET C QUESTION Which of these sets best shows the kind of pattern you would expect to see as the snowflakes build up Set A ES Set B Set O LI Sets A and B Each kind of pattern 1s as likely A9 13 Write a sentence which begins It is very likely that the Queen using your own words to finish it It is very likely that the Queen vereer sese 14 Wr te a sentence which begins It is unlikely that the Queen using your own words to finish it 6 It is unlikely that the Queen 15 Write a sentence which ends is something that happens by chance using your own words to start 1t oe PEE e dy CNS 1 something that happens by chance 16 For the following phrases tick all those which you think mean exactly the same as has a 50 50 chance of happening A It may happen oy it may not 8 It has an even chance of happening C It will happen 50 times out of 50 9 It can happen sametimes E It has an equal chance of happening or not EE E F It is very unlikely to happen 2 sx 9 10 17 Two six sided spinn lt are marked with 1 s and
33. e os both 57 2v RA SRA aa 24 25 Same odda MCA z 5 Ke i bt ke otar eg C keme Leno f 7 Compared to UA hance LO tote Met Cdi fed ha eg Have 2 mol ta a WE date Cork cor 540 rn 2 Mere B than Cdra bik no comparison made AE AS Bader TA Cin F Vta E ce holivem 5 gt S x mart e F 2 mone within Ec RE ada O in beft b PETS Same varend u tack fya Badi hay eyl ase f 58 xk tame B 12 ega Sama rae hance 3 ES 6 a E bett Awe Back ie Urum 17 mora 25 13 e e 1 cant AU 122 more ta Dd 21 mer Blade but moe Cute 2 chance 90 827 23 har 5 2 24 E ER MCS 0 Omk frotan te Ala 2 preset 3 More Bin H Leao B ve 6 5 mere W u H 6 Atar 7 diffrence belim bago 18 6J 3 baz ES 3 6 more LESS I2 4 n G af ete Leas W BSW u bh 59 MCG t4 OTHER i 6 Han hall 0 V7 hiso chance
34. entage of total variation attributable to each main effect is as follows Main Effect Expla ned Var ance AH2 Grade 44 School Year 14 Sex 0 6 The dominance of intelligence AH2 Grade is clear its influence being three times that of age School Year When MA was substituted for AH2 Grade a similar result was obtained 40 variance indicating that MA is almost as good a guide to Test performance as is AH2 Grade This is not surprising in view of the high correlation 72 between the two factors Source of Variation F Significance Main Effects 485 001 AH2 809 001 Year 271 001 Sex 107 001 2 Way Interact ons 1 74 015 AH2 Year 2 20 004 AH2 Sex 0 83 504 Year Sex 0 96 428 3 Way Interactions AH2 Year Sex 102 433 Table 5 Analysis of Variance for Test Score PROBABILITY CONCEPTS LEVELS In order to attempt to allocate each pupil unequivocally to a concept level Guttman Scalogram Analysis Refs 5 6 was undertaken The aim is to group together selected Concepts Test questions into subtests representing different levels and to allocate a passmark for each level To be at a particular level a pupil must pass each lower level and also pass the given level Guttman Scalogram Analysis is a method whereby the subtests and passmarks can be optimised The main outcome was a scale comprising three levels plus zero level indicating total failure as shown in Table 6 Unfortunately but not surprisingly many of the more unu
35. etian Stages are normally achieved for comparison of odds problems for three levels of general intelligence Facility 100 Lepel 1 Lerel 1 Level 1 Ledel 1 90 4 evel 1 Level Levgl 1 l Level 2 Level 1 Level 1 80 gt Levgl 2 Leyel 2 Levkl 1 SS Lel 3 70 gt 1 Ratio Algebra Graphs Decimals Measure Fractions Vectors Rotation Positive ment Relection Negative Nos Fig 2 Stage 1 Levels 3rd Yeir Sample Facility 80 70 4 Levet 2 Levkl 2 el 3 arek 2 Level 2 Level Vul 2 60 mE Level 2 els Lepel 4 50 Level 3 40 Ratio Algebra Graphs Decimale Measure Fractions Vectors Rotation Positive PRUBAHILITY ment Reflection Negative Facility sol level 2 Level 3 404 Level 3 level 5 Level 5 Leyel 3 Level Level 3 30 lt 1 level Level 4 Measure Fractions Vectors Rotation Positive PROBABILITY Reflection Negative 201 mario Algebra Graphs Decimals ment Nos Stage 3 Levels 3rd Year Sample tig 4 Facility 40 30 vel 4 TE 20 level 6 Ledel 4 Level 6 Level 4 Level 3 Level 5 eye Lefel 4 10 H Ratio Algebra Graphs Decimals Fractions Vectors Rotation Positive Reflection Negative Nos Fig 5 Stage 4 Levels 3rd Year Sampl Level 4 PROBABILITY 11 10 I 1 la 12 SUMMARY OF FINDINGS SUMMARY OF FINDINGS RELATING TO GCE O LEVEL MATHEMATICS MULTIPLE C
36. i e the evening out of proportions with increasing numbers of trials 38 a most important concept Those questions relating to this topic Qns 8 24 were not wholly satisfactory because of inevitable language problems The results and follow up interviews suggest that a lot of practial activity and discussion is needed to develop an appreciation of this concept in the majority of pupils FURTHER RESEARCH Much further research in this araa is called for Anyone interested in pursuing this on however small a scale is invited to write to David Green or to Professor A C Bajpai at CAMET TURTHER DETAILS Greater detail of this research may be obtained by consulting Ref 12 or from the author ACKNOWLEDGEMENTS My grateful thanks extends to all the following East Midlands Schools who so readily assisted with this work East Midland Regional Examinations Board University of London School Examinations Department Social Science Research Council Loughborough University of Technology Computer Centre Library and Administration Staff Project research assistants especially Mrs Margaret Tomlinson Members of the Department of Engineering Mathematics Loughborough University of Technology especially Reda Featherstone Gordon Bell and Francis Coon Professor C Bajpai Director of CAMET 15 10 11 12 REFERENCES HART K M 1980 HART K M ed 1981 HEIM A W WATTS K P amp SIMMONDS V 1974 HEIM A W WATTS K P
37. k ula s nx TAALA Amara unld dil e effe eee Un awha Black om jor ffereuces o she say ore la Sana ov Leve Black Ltb mdan b ur e NOTE TL inkinprebation ta acded referral tthe 7 akronabtferward mark 00 Valid Cox Jappe lt h Tov aain werg ka ake vay edad pasible ek to he Hu muat Speech a 501 Akel me 1 th Some d ma m wll do will da or says 53 MS GUIDANCE MS7 7 b Gei Valid pc be hand rare might ct 1 no chance impossible Likely v Vald la zo b Lamy very 93 Tundra sun deto wl ele Mihal poss unlikely imposable cotas 2h U net pora Lay dam Aard and faot rule Ane Ahatia impossible A fO doket Pe aHa amd esa vey hel and nel ove 5 be maud Chance ui Do net ge Ha buuf Up Us fon bts gureko Vald wl Clan too ec tro alert nue Es Seem he a sequence nuur geks ban ttue Os Avgthar 54 MARKING CODES WC SA O MIT zin 2 e 2 1 eB 20007 c x 2 2 0 e nlet A 200 8 Hur 2 5
38. lar response patterns with substantial increases in performance for Years 3 4 and 5 4 Items which merely require the counting concept are well done by all Years 5 Items requiring the ratio concept are very poorly done particularly below Year 4 6 Various strategies are employed by pupils in answering comparison of odds problems as judged by analysis of Question 6 responses Counting and differencing are often suggested but rarely used consistently Ratio is rarely mentioned but is used by very able especially older pupils 20 of sample got all Qn 6 correct 7 Items involving an appreciation of randomness the stability of frequencies and inference were particularly poorly done with little evident improvement with age 8 Although most pupils coped well with a combinatoric item at a concrete level Qn 26a few could proceed to a formal level item Qn 26c d 11 a b 9 10 11 12 13 14 15 16 17 18 Pupils verbal ability is often inadequate for accurately describing probabilistic situations Very often certainty and high probability are equated as are impossibility and low probability Pupils freely associate a 50 probabilit with 1 outcomes which might or might not happen 11 equi probable outcomes when more than 2 outcomes exist Generally boys consistently outscore girls on total Test Score but not on the Combinatoric questions see Table 4 Generally boys show superior perf
39. n je qe M do qe P o Y us pa L lu LA 1 juf jaja o vole 515 An CARD DETAILS Ead mod ie dat fo ong Subt recodid X cad CARD A 4 me Abe de E SU 7 A CARD B well dee Sam cda Tle cad T ge 255 z CARD Dedalo Gi eS eee pow The cado schalt geno Vere QAO 1290 cad ez _ CTO quads 1540 card VERZ GIO renne 1340 cad W hte mel OB ard Vs WO remde 20 cad 44 4 Yer Code AN S 6 8 Pigh Coda NNN 0 Le eet A EE 2120 Qu 2 s t 5230 d ds EE Qu Mason MONEO de 7 6 QE enen 13 4500 en iN K Qua o Me th Pi A 27 28 __ Eege n a ier en rc EA E NN SOT RA NEP 33 34 Qa 66 remo __ 36 Qu 7 Br 37 6 7 a n 38 76 60 57 8003060 ii to 0 0 7 4 oi 700 0 43 a 7 i ye 6 7 9 6 74 45 47 Qu 48 A 9 49 6 0 SO GA 26 0 EE 3558 Ce SOR
40. number or numbers 15 it hardest to throw or are they all the same ANSWER 5 An ordinary coin is tossed five times and Heads appears every time Tick the correct sentence below A Next time the coin is more likely to turn up Heads again B Next time the coin is more likely to turn up Tails C Next time Heads is as likely as Tails E D Don t know gt 6 Bag A has got in it 3 black counters and 1 white counter Bag B has got in t 2 black counters and 1 white counter See the diagram If you have to pick a black counter to win A x a pr ze and you must not look in the bag vhich bag should you choose to p ck from Tick the correct ansver B 7 90 el A Bag A gives a better chance to get SEU B Bag B gives a better chance to get black C Both bags give the same chance ES D Don t know Why dum SA de 5 GE 5 6 b Two other bags have each got some black counters and some white counters in them Bag C 5 black and 2 white Bag D 5 black and 3 white Which bag C or D gives a better chance of picking a black counter or do they give the same chance A Bag C mi seen C Same chance D Don t know Why ee en co 4 6 c 6 d 6 e A4 Two other bags also have black and white counters Bag E 2 black and 2 white E
41. ormance to girls on many Test questions with girls significantly superior on one question only Qn 20 General reasoning ability AH2 Grade is the dominant factor in Test Score 44 being three times as important as age School Year 14 Sex s of minor significance 3 The style of teaching along the teacher centred nd v dual sed learning continuum does not appear to be a significant factor affecting Test Score Generally boys score more highly than girls on Probability Concepts Level Probability Concept Level means rise consistently with increasing age School Year and intellectual level AH2 Grade Most pupils do not attain the level of formal operations Piagetian Stage III by Year 5 and presumably leave school at concrete operations level Stage II IMPLICATIONS AND RECOMMENDATIONS In studying the discussion which follows the reader is advised to refer to the Appendices and to Section 10 There is a great need for practical activity from an early age on which to build adequate experience for the pupil A scheme of guided discovery is called for This theme is reiterated later in this section The subject of probability is a very appropriate topic whereby to emphasise the nature of mathematical modelling Construction of Manipulation of Pbysical inter a mathematical the mathematical pretation of model model mathematical results Question 1 is asking the pupil what he thinks
42. re boys out of the first 100 babies born in a new hospital A They are equally likely B 7 or more out of 10 more likely E 1 C 70 or more out of 100 is more likely D No one can say 25 A bag has n t some white balls and some black balls A boy picks out a ball notes its colour and puts it back He then shakes the bag He does this four times He picks a black ball every time He then picks out one more ball What colour do you think he is likely to get 5 Tick the correct sentence A Black s more likely aga n 7 B Black and white are equally likely C White is more likely this time 15 31 26 a b c d Al6 Three boys are sent to the headmaster for steal ng They have to l ne up in a row outside the head s room and wait for their punishment No one wants to be first of course Suppose the boys are called Andy Barry and Christopher A B C for short We want to write down all the possible orders in which they could line up For example A B C we write ABC as shown below st 2nd 3rd ANSWER ABC A 2 wf es 5 an 5 EN EEN Now write down all the other different orders How many different ways are there altogether ANSWER H Now do the rest of the question If four boys A B C D have to be lined up how many different ways are there ANSWER And if
43. sual and interesting items in the Concepts Test were eliminated 6 Level Items n Level Pass Criterion o Failure at Level 1 1 3 4 5 26a 2 4 2 2 6b ansver 7 11 6b reason 6c answer 6c reason 6d answer 9 10 18 19 answer 19 reason 3 6d reason 6e answer 3 3 68 reason Le Table 6a Details of Probability Concepts Scale Coefficient of Reproduc b l ty Coefficient of Scalab l ty Non Scale cases Errors 2 Table 6b Stat st cal deta ls of Guttman Scalogram Analys s for Probab l ty Concepts Scale The distribution of Levels is shown in Table 7 Probability Concept Level Year Sex o 1 2 3 Mean Level 1 15 51 31 3 1 21 Girl 18 60 21 2 1 07 2 Boy 12 44 38 6 1 37 Girl 11 55 27 6 1 28 3 Boy 5 31 48 16 1 75 Girl 6 43 40 12 1 57 4 Boy 6 25 44 25 1 87 G rl 6 34 40 20 1 74 5 Boy 2 16 50 32 2 11 Girl 3 27 44 26 1 93 Table 7 Percentages of Main Sample Pup ls n each Concept Level by School Year and Sex A steady progression of Concept Level with increasing age is apparent and confirmed by the means for the five years Boys show superiority to girls consistently but the gap between them as judged by the mean Concept Level does not widen between the ages of 11 16 years It would appear that the difference is either innate or more plausibly inculcated in the early formative years Further analysis by intelligence level throws light on the nature of the d fference
44. t marbles 3 will get the most marbles E 22 1 and 2 will get a lot of marbles and 3 will get a few LJ None of these 1 2 3 Each channel will get about the same number of marbles 2 will get about twice as many marbles as 1 or 3 About half will go down 1 and about half down 3 A few will go down 1 nearly all down 2 and a few down 3 None of these 213 LI 29 22 23 A robot s put into a maze which t begins to explore Junction the robot is as likely to go down any one path as any 814 other except it does not go back the way it came traps at the ends of the eight paths see picture or traps is the robot most likely to finish up or are ill traps equally likely ANSWER There are eight In which trap A packet of 100 drawing pins was emptied out onto a table by a teacher Some drawing pins landed UP eb and some landed DOWN a The result was UP 68 DOWN 32 Then the teacher asks a girl to repeat the experiment Choose from the list below the result you think the girl will get A B C D E UP 36 UP 63 UP 51 UP 84 All these results have the same chance DOWN 64 DOWN 37 DOWN 49 DOWN 16 14 A15 24 Which of the following results is more likely 1 Getting 7 or more boys out of the first 10 babies born in a new hospital 2 Getting 70 or mo
45. there are five boys how many ways Don t try to write them all down ANSWER 5 16 32 APPENDIX B B TEST RESULTS ITEM RESPONSES indicates the response s considered to be correct QNI g z Year A B ES D Year A B D 1 18 28 45 9 1 5 38 53 4 2 15 24 55 6 2 4 43 51 2 3 11 20 66 2 3 4 61 35 0 4 12 13 71 4 4 2 60 37 1 5 11 12 74 4 5 4 71 25 0 All 14 20 61 5 All 4 53 42 2 QN3 a QN3 b Year A B D Year Area Counting Ratio Position Other Omit Concept Concept Concept or speed 1 4 71 22 3 2 4 76 16 3 1 28 28 2 13 19 10 3 2 87 11 1 2 33 34 2 9 16 8 4 3 87 9 1 3 39 33 6 7 9 7 5 3 92 5 0 4 39 30 7 3 7 12 All 3 82 13 2 5 39 28 9 5 6 13 All 35 31 4 8 13 10 qua Qus 6 Year 1 2 3 4 5 6 Same Other Other Omit Year A B cv D 1 1 0 0 0 0 23 67 2 1 5 1 14 14 67 5 2 20 22 69 2 1 3 2 10 14 73 3 3 1 0 13 79 4 1 2 3 11 10 78 1 4 2 0 0 0 0 14 79 3 1 0 4 10 10 78 1 5 00 9 86 2 1 1 5 10 9 80 1 All 1 0 0 0 0 17 75 3 1 2 All 11 12 75 2 ONG a QIN6 b Year A B C D Year A B C D 1 88 4 6 2 1 55 9 33 2 2 87 6 6 0 2 60 9 31 1 3 88 5 7 0 3 71 5 23 1 4 90 4 5 0 4 15 4 21 0 5 90 5 5 0 5 81 3 15 1 All 8B 5 6 1 All 67 6 25 1 33 gt B2 gn 61c Year B cx D Qn 6 9 Year A B E D 1 11 44 43 3 1 8 38 51 2 2 12 36 51 1 2 6 50 41 4 3 9 26 64 1 3 6 62 31 1 4 8 18 73 1 4 5 68 26 o 5 5 14 80 o 5 4 74 20 2 All 9 29 60 1 All 6 57 35 2 Qn 6 e Year A B D Qn 7 a 1 Year 1 2
46. to help pupils understanding and should be provided wherever possible or pupils encouraged to interpret verbal problems pictorially before proceeding to solve them Various strategies are employed by pupils in solving comparison of odds questions Qn 6 but little consistency can be found Much benefit can accrue from explicit classroom discussion of appropriate strategies for particular problems with a view to developing adequate discernment of types of problem or of mastery of a universally applicable strategy The verbal weakness of pupils has been exposed by questions 7 13 to 15 The teacher needs to give attention to educating pupils in the normally accepted usage of various terms which indicate varying degrees of likelihood Commonality of meaning is obviously most desirable and should not tacitly be assumed Discussion of events incorporating a chance element needs to be encouraged The idea of expectation Qn 9 seems to be fairly widely available despite its rarely being met in lessons It seems natural to capitalise on this important concept in introducing probability as a meaningful study of the real world In our scientifically and technologically dominated society with its deterministically oriented education system it is perhaps inevitable that the concept of randomness should be neglected Pupils responses to Qns 11 12 20 for example reflect this Very little time or attention is given to this concept despite the fact th
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