c s Online Interview

M a t h e m a t i c s Online Interview

Early Numeracy Interview Booklet Published by the Communications Division, for the Office of School Education, Department of Education, Employment and Training, GPO Box 4367, Melbourne, Vic. 3001, Australia. State of Victoria, August 2001 First published 2001 All rights reserved. Except under conditions described in the Copyright Act 1968 of Australia and subsequent amendments, no part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical and photocopying, recording or otherwise, without the prior permission of the copyright owner. Permission is hereby granted to the purchaser to reproduce appendices 1 and 2 from the book in quantities suitable for non-commercial use. National Library of Australia Cataloguing-in-Publication data Early numeracy: Bibliography ISBN 0 7594 0167 5 1. Numeracy Study and teaching (Primary). I. Victoria. Dept. of Education, Employment and Training. 372.72044

C ontents Section 1 Assessment 2 Assessing early numeracy 3 Value of the one-to-one interview 3 Stages of mathematical growth 4 Links between points of growth and the Interview 4 Section 2 Introduction to the Mathematics Online Interview 7 About the Interview 7 Equipment 7 Preparation 8 Recording responses 9 Before the first interview 10 Administration Functions 10 Determining starting points 11 Conducting the interview 12 Analysing the record sheet 15 Student profiles 16 Section 3 Mathematics Online Interview Questions 21 Glossary 51 Bibliography 53 Section 4 Appendix 1 Equipment checklist and photocopiable support materials 57 Appendix 2 Interview record sheet 93 Appendix 3 Points of growth 97

S E C T I O N 1 Assessment

A ssessing early numeracy The Early Numeracy Research Project (1999 2001) was commissioned by the Department of Education, Employment and Training and managed by the Early Years Branch. As part of the Early Numeracy Research Project (ENRP), a need was identified for the development of a comprehensive assessment tool for early numeracy. Through the Early Numeracy Research Project a framework of growth in numeracy learning was created, informed by available research on young children s mathematics learning. See Bibliography (p. 53) for references to research and related publications. The framework provided a means of tracking students learning through significant growth points in number, measurement and space, organised into the domains of Counting, Place value, Strategies for addition and subtraction, Strategies for multiplication and division, Time, Length, Mass, Properties of shape, and Visualisation and orientation. The points of growth identified through the research project were used as the basis for the stages of mathematical growth in the Early Years Numeracy Program. This framework was then used to develop appropriate hands-on assessment tasks where students could demonstrate mathematical understanding and preferred strategies for solving increasingly complex tasks. These tasks have been presented to more than 11000 students during the research project in the form of a one-to-one interview. Value of the one-to-one interview The Mathematics Online Interview (Section 3) is a powerful tool for assessing students numeracy development during the first five years of schooling. Evidence of student learning gathered during the interview allows teachers to: develop a profile of student numeracy development in the aspects of mathematics assessed in the interview inform focused teaching of students. Time spent conducting the one-to-one interview is invaluable in enhancing teachers understandings of an individual student s Anecdotal evidence and other data collected through ongoing monitoring and assessment of other mathematical areas can be used to support and expand the student profiles obtained through the interview. Interview booklet

Assessing Early Numeracy 1 The teachers in the Early Numeracy Research Project trial schools have found that the data from the interviews were revealing of student mathematical understanding and development in a way that would not be possible without that special opportunity for one-to-one interaction. (Clarke 2000) mathematical understandings and the strategies they use. For this reason it is recommended that the classroom teacher administer the interview. Stages of mathematical growth Stages have been outlined to illustrate likely mathematical growth of students in the early years of schooling. These should not be viewed as discrete, sequential stages of mathematics development. At any time, students will demonstrate a range of skills and understandings related to the mathematics in which they are engaged. The stages of mathematical growth: reflect the findings of relevant research in mathematics education from Australia and overseas allow mathematical profiles of individuals and groups to be developed form the basis of planning and focused teaching support the identification and description of student improvement enable the identification of students who may benefit from additional assistance in mathematics have sufficient scope to describe the knowledge and understanding of most students in the first five years of schooling. Within each mathematical strand, stages are described by points of growth. growth and the Interview Judgements based on one interview on a particular day need to be informed by ongoing assessment during mathematics sessions. Tasks in the Interview are linked to points of mathematical growth. Student responses from the Interview can be tracked against these points of growth to develop a profile of mathematical skills and understandings in the sections of the Interview conducted. The student profile provides a guide in establishing a mathematical focus for teaching. Interview booklet

S E C T I O N 2 Introduction to the Mathematics Online Interview

I ntroduction to the Mathematics Online Interview About the Interview The Interview includes oral questions that invite students to respond verbally or to demonstrate their level of understanding using a variety of materials and equipment. An example is provided opposite. SECTION E: Time 3 Early Numeracy Interview Equipment children s own hand-drawn clocks moveable yellow clock yellow June calendar card digital clock showing 12:51 excerpt from TV guide blank digital and analogue clockfaces Prior to the interview, children need to be asked in class to simply draw a clock. If they choose to draw a digital clock, please ask them to then draw an analogue one. They then bring these clocks to the interview. Equipment The equipment and cards needed for interview tasks are listed at the start of each section of the interview. The full list of required equipment is included as Appendix 1, along with photocopiable support materials. My Clock Some materials will need to be prepared before the interviews commence, for example, photocopying cards and shapes, colouring the squares on the number charts, cutting candles, and preparing the straw, string, paper clips and streamers for the length tasks. For ease of use, it is recommended that schools use the colours indicated for each set of cards and shapes that need to be copied, as they are referred to by colour in the interview instructions. The interview will be easier to conduct if all materials are arranged on a separate table in the order in which they will be used. Tasks for each section of the interview can be stored together and clearly labelled for easy access. 38 Depending upon the kinds of information shown in the child s drawing of the clock, ask questions like the following: Tell me about your clock. What are clocks for? What are the numbers on your clock? (If relevant.) How do the numbers work? What time does your clock show? Tell me what you do at this time. 39 Telling the Time Tell me what time these clocks show? (use the yellow clock faces) [at any stage] a) 2:00 b) 9:30 c) 2:20 EARLY NUMERACY Interview booklet Q40 3 Interview booklet

Introduction to the Mathematics Online Interview 2 Preparation For more information see the School User Guide. Before conducting interviews, teachers need to be familiar with the tasks, the wording to be used, appropriate starting points for their students, the possible responses students might give and the different pathways that can be taken after each task. It is useful to practise recording student responses. As a model for how this is done, it is worth watching an experienced colleague conducting an interview. A comfortable environment is needed for the interview. It is recommended that the teacher and student sit on the same side of the table on similar-sized chairs. Right-handed teachers will find it preferable for the student to sit on their left. The teacher can then write on the record sheet away from the student s workspace. This positioning is reversed for left-handed teachers. The photocopiable record sheet is included as Appendix 2 of this booklet. Importing existing records This function is accessed by pressing the Import button on the main page. Import is used in order to add students to the new online database by importing them from the old version of the Interview. Students and their interview records are first exported to an XML file from the old Interview, and then imported into the new Interview via the Import function. Students imported are assigned to the nominated class, and their interview records are assigned to the nominated session. Interview booklet

2 Introduction to the Mathematics Online Interview Before the first interview Teachers must have a current Edumail account to access this program. When teachers log into the online Interview for the first time, they may find that the program is empty. Details of classes, students and groups may not have been entered, and interview records will not exist on the system. If there are no classes in the system teachers cannot proceed. Before teachers can start using the Interview, the School Administrator will need to: import existing interview records from school-based servers create classes to which the students will be added add students to the classes assign teachers to the classes. Only School Administrators are able to create classes for the school, and to add students and teachers to those classes. Once a teacher has been assigned to one (or more) classes containing students, they will be able to view these classes, and interview those students. It is important to understand that all students and classes for all schools will have their interview information stored in the same centralised place. For this reason, it would be impractical for all teachers to be able to see lists of all students and classes in the system. Rather, when teachers log on to the Interview, details of the classes shown in the class list, and of the students shown when a teacher selects a class, are restricted to those students and classes to which that teacher is linked. Interview booklet

Introduction to the Mathematics Online Interview 2 Administration Functions Users who have Administrator-level access will be able to access the Interview s Administrator functionality via the Admin button on the main screen. There are four main Administrator functions. Manage Classes Manage Students Bulk Add Students User Management Assigning a School Administrator Schools must assign a School Administrator before any interviewing can commence. Principals automatically have School Administrator access. The Principal should access the User Management function and delegate School Administrator access to another user at the school. This person will usually be the Numeracy Coordinator. School Administrators may access all of the School Administrator functionality within the Interview, such as Manage Classes, Manage Students and Bulk Add Students. They are also allowed to run Profiles for the whole school, as opposed to being restricted to students in their classes. To assign School Administrator access, press Admin and then press User Management and you will be presented with a screen for adding or removing School Administrator users. School Administrator access is reserved for a few Interview users who need access to all aspects of the Interview. For more information see the School User Guide. 10 Interview booklet

Determining starting points Skip count It is recommended that the interview be conducted with Count all by 1s all students starting school and again at the beginning of each new school year. Students in their first year at school will need to 31. Teddies at the movies (circle strategy used) commence with Task 1 and should Answer also attempt the tasks in Skip count the First year detour. Older students who are new to the Count by 1s school could start with these tasks, unless it is obvious that 32. Multiplication problems a. 3 x 10 d. 3 x 50 their understanding is beyond the content at this level. Starting points for students (other a. 16 2 than those in d. Prep) 24 3 c f. b. 60 10 e. 35 5 should be informed by ongoing monitoring and assessment. It is c. 80 4 f. 35 7 34. Off to the circus suggested that the interview begin with a task that is just below Answer 35. Sharing our money the student s current performance level to ensure initial success. If Answer 36. In your head (23 x 4) difficulties are experienced, the interviewer should move back to the Answer Strategy previous task/s. 38. My clock Recording responses Awareness: State of Victoria, Department of Education, Employment and Training, 2001 Other 29. Tennis balls (circle strategy used) a,b. Answer Skip count 2 Known fact Count all by 1s Other c. Answer Skip count Known fact Count all by 1s Other 30. Dots array a,b. Answer Known fact Other c. Answer Skip count Known fact Count all by 1s Other Known fact Other b. 2 x 7 e. 4 x 30 c. 10 x 7 f. 5 x 7 33. Division problems 37. Missing number (54 x = 2) a,b. Answer(s) Strategy c. Answer(s) Strategy E. TIME (at least) 1 feature: (at least) 1 purpose: Recording a student s response varies according to the task. The majority of tasks have a box where a correct response is indicated: with a tick If a task is not used or not answered correctly with the student, the box is left blank Some tasks require detail about the student s response. This might be the number the student counted to, the strategy used, or the explanation of the student s thinking. It is recommended that teachers record as much as possible of students responses to gain a deeper understanding of the mathematical thinking in the comments box at the bottom of each question page. *Do not record comments in the answer field. e. Month before April 41. Calendar a. 18th of June b. Day of week c. Last day of June d. Month after June e. Day of July 1 42. Duration tasks a. 13 minutes after 12:51 b. 13:04 changed to 1:04 43. TV guide Movie length 44. Linking digital and analogue a. 8 to 5: Analogue Digital F. LENGTH MEASUREMENT 45. String and stick comparison a. Prediction b. Accurate comparison c. Answer Some awareness of the concept of length (as indicated by responses to whole question) 46. Long straw a. Correct use of non-standard units b. Answer (including units) 47. Using the ruler a. Correct use of ruler b. Correct answer including units 48. Tearing the streamer a. Student s torn piece (range 75 cm to 125 cm) b. Student s stated measure of 93 cm piece c. Stated difference Introduction to the Mathematics Online Interview G. MASS MEASUREMENT 49. What do you notice? Accurate comparison Some awareness of mass (e.g., light, heavy ) as indicated by responses to whole question 50. Teddies and coins Estimate (if any) Answer Method 51. One kilogram a. Prediction b. Answer Method 52. Using formal units Correct use of formal units Answer 53. Using kitchen scales a. Correct use of scale Answer (including units) b. Correct use of scale c. Explanation d. Answer EARLY NUMERACY Interview booklet c. Answer Skip count Known fact Count all by 1s Other 28. Sharing teddies (circle strategy used) Answer Share by groups Known fact Share by 1s Other 29. Tennis balls (circle strategy used) a,b. Answer Skip count Known fact Count all by 1s Other c. Answer Skip count Known fact Count all by 1s Other 30. Dots array a,b. Answer Skip count Known fact Count all by 1s Other c. Answer Skip count Known fact Count all by 1s Other 31. Teddies at the movies (circle strategy used) Answer Skip count Known fact Count by 1s Other 32. Multiplication problems a. 3 x 10 d. 3 x 50 b. 2 x 7 e. 4 x 30 c. 10 x 7 f. 5 x 7 33. Division problems Interview booklet a. 16 2 d. 24 11 3 b. 60 10 e. 35 5 c. 80 4 f. 35 7 34. Off to the circus ment and Training, 2001 5 4 App Know Know Movi Some (as in Some as ind 39. Telli a. 2:00 b. 9:30 c. 2:20 40. The a. All da b. All m d. Day b e. Mont 41. Cale a. 18th b. Day o c. Last d d. Mont e. Day o 42. Dura a. 13 m b. 13:04 43. TV g 44. Link a. 8 to 5 F. LENG 45. Strin a. Predi b. Accu c. Answ 46. Long a. Corre b. Answ 47. Usin a. Corre b. Corre 48. Tear a. Stude b. Stude c. State G. MAS 49. Wha c f. Accu 50. Tedd

Introduction to the Mathematics Online Interview 2 Entering comments in Numeric Fields Numeric fields such as 1st difficulty (see example below) need to be handled properly, or the input can be misinterpreted. There are a number of growth points which evaluate numeric input in order to assess a result. Remember Only insert a single number into these numeric fields. If more figures are entered into these fields, profiles may produce and error or incorrect reporting data. If teachers choose to record further observational notes, please do so in the Comment field at the bottom of each question page. For example: This is an invalid entry. This numeric field 2a contains 13-16 which happens to be a valid mathematical statement = -3. This answer -3 will be evaluated by the program and produce an unpredictable result. 12 Interview booklet

2 Introduction to the Mathematics Online Interview Selected Session The Mathematics Online Interview builds on one Interview per student. It is critical that teachers select the correct session when continuing to interview a student and update records. When interview records are updated, they are assigned to a session, e.g. Grade 2 Semester 1. This way, when viewing profiles, teachers can elect to restrict the results to include only answers up to and including a chosen session. This will enable teachers to differentiate between the achievement of growth points over time. Ongoing updates throughout the first six months of the year should be assigned to Semester 1 sessions, and any updates throughout the second part of the year should be assigned to Semester 2 sessions. What if you import to the wrong session, or interview a student and realise that you ve selected the wrong session? ANSWER: Teachers will need to re-visit each of the questions the students attempted. Insert changed session to into the comments field. Because the comment has been changed, the question will be updated, and the session will be fixed in the process. Conducting the interview The aim of the interview is to determine the most sophisticated strategies a student chooses to use in each of the mathematical areas assessed. The entire Interview takes between 30 and 40 minutes to conduct with each student providing rich information on the areas of mathematics covered. Initially the interview may take longer until the interviewer is familiar with tasks. Before their interview commences, students should be reassured that this is a special time in which the teacher will find out more about how they think while they are doing mathematics tasks. It should be explained that the teacher will follow the order of the tasks & record what happens using their Laptop. The wording of questions has been finely tuned during the Early Numeracy Research Project. It is important that interviewers use the exact words in the interview when introducing each task, as this enhances the consistency of the data collected and avoids misinterpretation. Interview booklet 13

Introduction to the Mathematics Online Interview 2 This is an opportunity for listening to students rather than prompting or teaching and should be a positive experience for all involved. When asking each question, the interviewer should: pause to allow time for the student to interpret the question and construct a response pause after the student s response in case an elaboration or correction is given be aware of non-verbal messages that might be given be accepting of the response without implying whether it was correct or not encourage the student s efforts, focusing on behaviour rather than the responses. When conducting the interview with students for whom English is a second language, the following options are suggested: the tasks can be presented in English using the wording in this booklet so the level of mathematical language and concept understanding in English can be determined an interpreter could be used so that the focus is on the student s mathematical understanding in their strongest language (when conducting the interview with an interpreter or teacher aide the teacher should ask the questions, which are then interpreted). The full interview involves around sixty tasks, some with sub-tasks, however no student will attempt all of these. The pathway taken through the interview is determined by each student s responses. At the end of each task, the interviewer makes one of three decisions: 1 While the student is successful, the interviewer continues to the next task in that section. 2 If the student has difficulty with the task, the interviewer moves into a detour designed to elaborate more clearly the student s level of understanding in the particular content area. 3 If there is no detour, the interviewer abandons that section of the interview and moves on to the next section. When a task is complete, the interviewer ensures that all responses have been recorded, then may ask the student to pack up the materials or cards. This allows time for organising and reading the next question. 14 Interview booklet

2 Introduction to the Mathematics Online Interview There is a range of symbols and styles for the text used in the interview: sentences in italics are instructions to the interviewer and sometimes include extra advice about the task sentences in normal type are the words to be said to the student means follow the arrow if there is a tick in this box means follow the arrow if there is a incorrect response/left box blank if there is no instruction to the contrary, continue to move down the page. The interview continues until the student has difficulties with a task in the final section assessed. Students should be praised so that they feel positive about the experience. Analysing the result The information recorded during the interview can be used to determine the points of growth a student is currently demonstrating. Achievement of some points of growth will be indicated by success in just one or two tasks, but achievement of others will require successful demonstration across more tasks. For example in Question 18: Counting On, the student is asked to find the total of two collections of teddies (nine and another four). Some students will count all to find the total (1, 2, 3,..., 11, 12, 13), some others will realise that by starting at 9 and counting on (10, 11, 12, 13), they can solve the problem more easily. For these students the response will indicate where the student is currently operating, that is: Count all to find the total of two collections, Count on from one number to find the total of two collections, or Neither. Question 21: Basic Strategies requires successful completion of a number of tasks. The student is asked simple addition and subtraction questions. All five questions need to be answered correctly, with at least three preferred strategies for solving them being demonstrated (these strategies are highlighted on the record sheet). Success in these tasks will indicate that the student is achieving at these points of growth in Number, that is: Interview booklet 15

Introduction to the Mathematics Online Interview 2 In addition or subtraction problems, use strategies such as doubles, commutativity, adding 10, tens facts, and other known facts. The student responses can be used to determine which specific points of growth students have demonstrated. Interpretation of student responses, and which points of growth they demonstrate, can form the focus of meetings of professional learning teams to develop a shared understanding of the strategies students are using. This discussion would also enhance the consistency of the data being collected. Student profiles A student profile can be developed, based on analysis of the content of the interview. Student Number Measurement Shape Amy Knows some number names, but has difficulty stating them in sequence above 10. Has awareness of the attribute of length and its descriptive language (longer, shorter). Can sort simple shapes. Can rote count the number sequence to 10, but is unable to reliably count a collection of that size. Can read and record some single-digit numbers (confuses 6 and 9). Has awareness of the attribute of mass and its descriptive language (heavy, light). Can describe at least one feature (numbers and hands) and one purpose of clock faces (tell the time). Running a profile interprets the Interview answers for one or more students, and compiles the results into a report. To generate a profile, press the Profiles button on the main page. The Mathematics Online Interview supports the following Profiles. 16 Interview booklet

2 Introduction to the Mathematics Online Interview Student Profile This profile shows a per-student detailed summary, including a summary of the growth points achieved for each student. Note that in the old version, this profile was roughly aligned to the CSF levels. This function has been removed with the introduction of the Victorian Essential Learning Standards. Group Profile This profile shows a per-student or a selected group profile of which growth points were achieved, and when. Profiles are grouped according to the defined groups, including Number, Space and Measurement. New to the Interview Profiles can also be printed in more detailed groupings such as Counting, Addition & Subtraction, Multiplication & Division and Place Value. Question Summary This profile is an aggregate report. It shows how many students in the selected group have answered each question correctly. Points of Growth Summary This profile is an aggregate report. It shows how many students in the selected group have achieved each growth point. Question Summary for a Student New to the Online Interview This profile is a per-student report showing which questions have been answered correctly, and when they were attempted, if more than once. Where questions have been answered multiple times, this profile indicates the number of times the question has been attempted. Clicking on the No. of Attempts link shows an audit of responses, and when they were made. Interview booklet 17

Introduction to the Mathematics Online Interview 2 This report may be run for multiple students, but it is intended to be produced for one student at a time, and it takes some time to run. The Mathematics Online Interview is only one piece of assessment that may be utilised to illustrate students development in Mathematics. This assessment, along with other assessments tools, will enable teachers to make a holistic, on-balance judgement against the standards. It is generally the case that a single assessment instrument will not tell us all we need to know about student achievement and how it can be improved. We therefore need to be familiar with and use a variety of assessment tools so we can match them closely to the type of information we seek. For more information see the School User Guide. 18 Interview booklet

S E C T I O N 3 Mathematics Online Interview Questions

Italics are for instructions to the interviewer normal type are for the words interviewer says record: tick box for a correct response leave blank for an incorrect response student response in box provided. 3 means that if you have put a tick in a box for this item, go to where the arrow directs. means that if you have an incorrect response/left box blank, go to where the arrow directs. If there is no instruction to the contrary, move down the page. Don t follow an instruction until you get to it (i.e. don t look ahead and act upon particular instructions until that point has been reached). Equipment for each section is listed at the beginning of that section. Many questions require the interviewer to select the strategy used by the child and tick the appropriate box. Interview booklet 21

Mathematics Online Interview 3 SECTION A: Counting Equipment ice cream container of teddies small plastic cup (which will hold at least 20 teddies) small, sealed plastic bag with collection of coins in it (1 x $1; 1 x 50 cents; 3 x 20 cents; 5 x 10 cents; 5 x 5 cents: total $2.85) 1 Teddy Task Show the child the teddies and get the cup. Cup must be able to hold more than 20. Please take a big scoop of teddies.... Please put a few more teddies in to fill up the cup (at least 20). a) Hold them in front of you.... Tell me how many teddies you think are in the cup. b) Please check to find out. Q2 If the child is in the first year of school or had difficulty counting the teddies, move to the detour on page 24 now, otherwise continue on. 2 Counting Forwards, Backwards, and Breaking the Sequence Please count for me by ones without the teddies. Start counting from I ll tell you when to stop. a) 1... 32 part (e) b) 53... 62 part (d) c) 84... 113 (remember to note first difficulty) d) Count backwards from 24. I ll tell you when to stop (24, 23,..., 15). (If child hesitates, say like 24, [pause] 23,... ) Q3 part (e) e) Count backwards from 10. I ll tell you when to stop. Section B (If child hesitates, say like 10, [pause] 9,... ) 22 Interview booklet

3 Mathematics Online Interview 3 Before and After Tasks a) When you are counting forwards, tell me the number that comes after 56. b) Tell me the number that comes before 56. 4 Counting from 0 by 10s, 5s, and 2s Count by 10s for me (then 5s, 2s) as far as you can go. [at any stage] (Stop child after 110, 45 and 30, respectively.) B 5 Counting from x by 10s and 5s a) Start at 23 and count by 10s. (Stop the child after 103.) B b) Start at 24 and count by 5s. (Stop the child after 44.) B 6 Counting from x by a single digit number a) Start at 11 and count by 3s. (Stop the child after 35.) B b) Start at 20 and count by 7s. (Stop the child after 55.) B 7 Counting Money Tip out the collection of money onto the table. a) Please count the money for me. (No recording allowed for this task.) b) What did you find? B c) How much more money would you need to have $5? (Any counting method yielding the correct answer is acceptable.) Section B Interview booklet 23

Mathematics Online Interview 3 Detour for children starting the first year of school (Also recommended for Year 1 and 2 children who had difficulty counting the collection in Q1) (Conservation of number; one-to-one correspondence; more/less; pattern recognition; ordinal number; language of location; subitising; simple counting, number before and after) Equipment a collection of 40 teddies of mixed colours in a container (plus a separated set of 20 with 4 yellow teddies, 5 red teddies, 3 green teddies, and 8 blue teddies), six pink dot flashcards including the blank card small pink cards with digits 0 9 five plastic cups nine straws four candles (all 2 cm in diameter; 5 cm, 10 cm, 15 cm and 20 cm in length) I Simpler Counting Tasks / More or Less / Conservation Place a pile of 20 teddies in front of the child in a scattered pattern, made up of exactly 4 yellow teddies, 5 red teddies, 3 green teddies, and 8 blue ones. a) Please put the yellow teddies together. b) How many yellow teddies are there? Put a group of 3 green teddies together near the 4 yellow teddies (two different small groups). c) Are there more green teddies or more yellow teddies? Push the yellow and green teddies aside. d) Please get five blue teddies. e) Now put them in a line. (If the child has already put them in a line, ask the child to move them together now.)... Tell me how many blue teddies there are. II Location / Pattern / Ordinal Number a) Please put out a yellow teddy.... Now put a blue one beside it.... Now put a green one behind the blue teddy.... Now put the green teddy in front of the blue teddy.... Now watch what I do with the teddies. Make a pattern with the teddies (G, Y, B, B, G, Y, B, B) in front of the child. b) I ve made a pattern with the teddies. Please say the colours for me as I point. 24 Interview booklet

3 Mathematics Online Interview Hand the container of teddies to the child. c) Please make the same pattern. d) (If the child s pattern is a correct copy, point to it. If not, point to your pattern.) Please make the pattern go on a bit more. e) How did you decide what came next in the pattern each time? f) Point to the green teddy in 1st position. The green one is the 1st teddy in my pattern. You point to the 3rd one. What colour is the 3rd teddy? You point to the 5th teddy. What colour is the 5th teddy? III Subitising / Matching Numerals to Quantities / Ordering / One-to-one Correspondence / Part-part-whole a) I m going to show you some cards quite quickly. Tell me how many dots you see. Show each pink flashcard for 2 seconds only, in the following order and orientation: Now put the dot cards all down in the order shown above. Spread out the pink 0 9 cards randomly, face up, in front of the child, between the child and the dot cards. 8 5 1 6 4 2 0 9 7 3 b) Find the number to match the dots. (If the child seems puzzled that there are more numeral cards than sets of dots, explain that you won t need to use all the numbers.) Remove the dot cards and the zero card. Shuffle the numeral cards and place them face up randomly on the table. c) Please put the number cards in order from smallest to largest. If the child is successful, hand across the zero card. d) Where would this one go? e) Please show me 6 fingers.... (If correct... Can you show me 6 fingers another way? Another way?) Interview booklet 25

Mathematics Online Interview 3 f) When you are counting by ones, what is the number after 4? (If successful... 10? If successful... 15?) g) What is the number before 3? (If successful... 12? If successful... 20?) Place 5 cups out in a line. Hand the child 9 straws. h) Please put one straw in each cup. Spread out three candles (20 cm, 5 cm, and 10 cm in that order from left to right). i) Please put these candles in order from smallest to largest.... Please point to the largest.... Please point to the smallest. (If successful, add in the 15 cm candle. This time, place the candles like this: 10 cm, 20 cm, 5 cm, and 15 cm, in that order from left to right.) j) Now put these candles in order from smallest to largest.... Please point to the largest.... Please point to the smallest. Go to Question 2, page 22 (for all children) 26 Interview booklet

3 Mathematics Online Interview SECTION B: Place Value Equipment teddies green number cards mauve cards with digits 0 to 9 calculator 2-digit chart 3-digit chart blue, yellow, pink and orange card sets icy pole sticks (including eight bundles of ten and twenty loose icy pole sticks) white 36 card white cards with 2791 and 3027 respectively pink chart with Australian city populations orange page of number lines 8 Reading Numerals Show the child number cards (green) one at a time. Discontinue at first difficulty. a) Read these numbers: 3, 8, 36, 83, 18, 147, 407, 1847. If the child has difficulty with any of 3, 8, 36, or 83, spread out the mauve cards with digits 0 to 9 face down on the table. b) Pick a card and tell me the number you have taken. After moving through the whole set, point to the 7 card. c) Get me this many teddies. Any trouble with 1-digit numbers in Q8: Q10 9 Calculator Tasks (Hand the child the calculator.) Have you used a calculator before? Please turn it on. a) Type these numbers on the calculator (7, 47, 60, 15, 724, 105, 2469, 6023) (Stop when the child is not successful. Ask the child to clear the calculator between numbers.) Interview booklet 27

Mathematics Online Interview 3 b) Step 1: Pick any number from 2 to 9, and type it on the calculator. Read the number. (Don t clear the calculator.) Step 2: Type in a different number from 2 to 9 (forming a 2-digit number). Read the number. Step 3: Type in a different number from 2 to 9 (giving a 3-digit number). Read the number. [Continue Step 4, Step 5, etc. until first difficulty] 10 Ordering Task Based on the child s success in Q8 and Q9, select the 1-, 2-, 3-, or 4-digit set of cards Pick the number of digits for which the child has been completely successful to that point. [Move up or down through the card sets depending upon success; e.g., success with 2 digits, try 3; if unsuccessful with 2, try 1-digit cards, etc. If child has had earlier difficulty with reading 1-digit numbers, there is little point in continuing the ordering task.] Spread one set of cards out on the table. Do not read the numbers aloud. a) 1-digit set (blue: 2, 5, 9) b) 2-digit set (yellow: 19, 36, 74) c) 3-digit set (pink: 97, 156, 403, 813) d) 4-digit set (orange: 3569, 3659, 3956) Here are some numbers. Order these numbers from smallest to largest.... Please point to the largest.... Please point to the smallest. (The child does not need to read the numbers, only point to them.) in Q10a or Q10b Section C 36 19 74 11 Bundling Tasks Ask the child to unpack the icy pole sticks. Here are some icy pole sticks in bundles of ten. (Offer the chance to check a bundle if it seems appropriate.) Here are some more loose ones. Show white card for 36. 28 Interview booklet

3 Mathematics Online Interview a) Get me this many (icy pole) sticks. (If child starts to count all in ones, interrupt and ask them if they can do it a quicker way with the bundles. If they can t, C.) b) Tell me how you worked that out. C 12 2-Digit Chart Task Show the child the mauve 2-digit chart card. Look at the way the numbers go. a) Tell me which number goes in the blue square. b) Please explain. C 13 3-Digit Chart Task Show the child the white 3-digit chart card. This is a different chart. Look at how these numbers go. a) Tell me which number goes in the orange square. b) Please explain. C Only continue on now if the child has had complete success on all tasks involving 1, 2, 3, and 4 digit numbers in Q8 Q13. Otherwise, go to Section C. 14 Ten More Show the child the white 2791 card. Pause for a couple of seconds for the child to look at the number. Tell me the number that is ten more than this number (2801). 15 One Hundred Less Show the child the white 3027 card. Pause for a couple of seconds for the child to look at the number. Tell me the number that is 100 less than this number (2927). Interview booklet 29

Mathematics Online Interview 3 Sorting the Capital Cities 16 Capital City Population (June 1998) Show the child the pink chart with population figures. Here is a list of Australia s capital cities (point to the names of the cities).... These numbers show how many people live in each city. [at any stage] C a) (point to the word Darwin) How many people live in Darwin? b) (point to the word Canberra) How many people live in Canberra? c) (point to the word Adelaide) How many people live in Adelaide? d) Please point to the city that has the third largest number of people. e) How did you work that out? Adelaide 1 088 349 Brisbane Canberra 1 57 615 308 086 Darwin Hobart 86 576 194 974 Melbourne 3 371 308 Perth 1 341 914 Sydney 3 986 723 17 Interpreting the Number Line Show the child the orange page of number lines. Point to the first one, pointing to the relevant numbers as you read the question. [at any stage] C a) The numbers on this line go from zero to 100... (pointing to the little mark). Round about what number would this be? (acceptable range: 55 to 75). b) The numbers on this line go from zero to 2000... (pointing to the little mark). Round about what number would this be? (acceptable range: 400 to 600). c) The numbers on this line go from 39 to 172... (pointing to the little mark). Round about what number would this be? (acceptable range: 65 to 95). d) The numbers on this line go from zero to one million... (pointing to the little mark). Round about what number would this be? (acceptable range: 700 000 to 800 000). Section C 30 Interview booklet

3 Mathematics Online Interview SECTION C: Strategies for Addition and Subtraction Equipment green teddies ice-cream lid white, blue and yellow problem cards 18 Counting On Place 9 green teddies on the table. a) Please get four green teddies for me. b) I have nine green teddies here (show the child the nine teddies, and then screen the nine teddies with the ice-cream lid). That s nine teddies hiding here and four teddies here (point to the groups). c) Tell me how many teddies we have altogether... Please explain how you worked it out. Q19 If incorrect answer, ask part (d). d) (Remove the lid). Please tell me how many there are altogether. 19 Counting Back For this question you need to listen to a story. a) Imagine you have 8 little biscuits in your play lunch and you eat 3. How many do you have left?... How did you work that out? If incorrect answer, ask part (b): b) Could you use your fingers to help you to work it out? (it s acceptable to repeat the question, but no further prompts should be given). Section D Interview booklet 31

Mathematics Online Interview 3 20 Counting Down To / Counting Up From I have 12 strawberries and I eat 9. How many are left?... Please explain. or not using a highlighted strategy in Q 20: D 21 Basic Strategies I am going to ask you some questions. Establish (if not known) what the child prefers (e.g., do you say 4 plus 4 or do you say 4 and 4? ) Please tell me the answer. Use clues such as speed of response to decide which strategy is used. [at any stage] a) 4 + 4 b) 2 + 19 c) 4 + 6 d) 27 + 10 e) 10-7 D Five and at least three highlighted strategies: Q22. Otherwise: Section D 22 Derived Strategies Here are some more questions. Please tell me the answers. Use clues such as speed of response to evaluate each strategy. [at any stage] a) 12 6 b) 7 + 8 c) 19 15 d) 16 + 5 e) 36 + 9 D Five and at least three highlighted strategies: Q23. Otherwise: Section D 32 Interview booklet

3 Mathematics Online Interview 23 Multi-digit Strategies I am going to show you some questions. Tell me the answer. Show the white cards for the following questions [at any stage] a) 68 + 32 b) 25 + 99 c) 100 68 For the final two (d and e), read the questions (no cards provided). d) half of 30 e) double 26 D 24 How Many Digits? Show the blue card with 134 + 689. Please read the card to me. a) Is the answer to this more than 1000 or less than 1000? b) Please explain. D Show the blue card with 1246 358. Please read the card to me. c) Is the answer to this more than 1000 or less than 1000? d) Please explain. D 25 Estimating and Calculating Addition Show the yellow card with 347 + 589. a) Please read this to me. b) Please estimate the answer to this (If necessary, prompt: what would the answer be round about? ) No estimate or one outside the range 800 1000 Section D Interview booklet 33

Mathematics Online Interview 3 c) Can you work out the exact answer to this in your head? (936) If yes (unlikely!), encourage the child to try to do so. If not successful (or if the response to the question in part (c) is no ), make the following request: Please use the paper to work it out any way you like. D 26 Estimating and Calculating Subtraction Show the yellow card with 642 376. a) Please read this to me. b) Please estimate the answer to this (If necessary, prompt: what would the answer be round about? ) No estimate or one outside the range 200 300 Section D c) Can you work out the exact answer to this in your head? (266) If yes (unlikely!), encourage the child to try to do so. If not successful (or if the response to the previous question is no ), make the following request: Please use the paper to work it out any way you like. Section D 34 Early Numeracy

3 Mathematics Online Interview SECTION D: Strategies for Multiplication and Division Equipment four empty matchboxes ( teddy cars ) orange page with teddy mats 12 teddies of the same colour plastic transparent container with three tennis balls inside it red card with dots cover card for dots pink and orange cards with multiplication and division tasks pink $52 card 27 Teddy Cars Put four matchboxes in a line. Here are four teddy cars. Please put two teddies in each car. a) How many teddies is that altogether? b) Tell me how you worked that out. c) If the child appears to be counting all, ask: Could you do that another way, without counting them one by one? 28 Sharing Teddies on the Mats Show the child the orange picture of four teddy mats. Put out 12 teddies of the same colour. a) Here are four teddy mats. Here are 12 teddies. Share the 12 teddies between the four mats, so that there is the same number of teddies on each mat. How many teddies go on each mat? b) How did you work that out? in Q27 or Q28 Section E Interview booklet 35

Mathematics Online Interview 3 29 Tennis Balls Task Put out 1 packet of 3 tennis balls. Here is a packet of tennis balls. a) How many balls would there be in four packets? b) Tell me how you worked that out. c) If the child appears to be counting all, ask: Could you do that another way, without counting them one by one? 30 Dots Array Task Here are some dots. Show red card (4 x 5) for an instant, in the orientation shown here. I m going to hide some. Cover the bottom 4 x 3 section, and the bottom half of the 3 dots above it. a) How many dots are there altogether on the whole card? b) How did you work that out? c) If the child appears to be counting all, ask: Could you do that a faster way, without counting them one by one? Unsuccessful in Q30 or points to hidden dots, counting them one by one: Section E 31 Teddies at the Movies Here comes another story. 15 teddies are sitting in rows at the movies. The teddies are sitting in three equal rows. How many teddies are in each row? How did you work that out? E 36 Interview booklet

3 Mathematics Online Interview 32 Multiplication Problems Show the child the pink card 3 x 10. Establish (if not known) what the child prefers (e.g., do you say 3 times 10 or do you say three 10s? ). Remove the card. Tell me the answers to these questions. Read the problems one at a time. [at any stage] a) 3 x 10 b) 2 x 7 c) 10 x 7 d) 3 x 50 e) 4 x 30 f ) 5 x 7 E 33 Division Problems Show the child the pink card 16 2. Establish (if not known) what the child prefers (e.g., Do you say 16 divided by 2? or do you say 16, how many twos? or do you say how many twos in 16? ). Remove the card. Tell me the answers to these questions. Read the problems one at a time. [at any stage] a) 16 2 b) 60 10 c) 80 4 d) 24 3 e) 35 5 f) 35 7 E 34 Off to the Circus 97 people are going to the circus. 20 people can ride in each bus. How many buses will be needed to get all 97 people to the circus? or physical modelling needed Section E Interview booklet 37

Mathematics Online Interview 3 35 Sharing our Money (pen and paper methods are acceptable for this task) Show the child the pink card ($52). Share $52 evenly between four people. How much does each person get? E 36 In Your Head Show the child the orange card with the expression: 23 x 4. a) Tell me the answer for 23 x 4. E b) Please explain. 37 Missing Number Show the orange card with 54 x = 2 a) The answer to 54 x? ends in 2. What can you tell me about this missing number? (Pointing to the space after the multiplication sign.) E b) How did you work that out? c) Could it be any other number? How do you know? Section E 38 Interview booklet

3 Mathematics Online Interview SECTION E: Time Equipment children s own hand-drawn clocks moveable yellow clock yellow June calendar card digital clock showing 12:51 excerpt from TV guide blank digital and analogue clockfaces Prior to the interview, ask children to simply draw a clock. If they draw a digital clock, ask them to then draw an analogue one. They bring these clocks to the interview. 38 My Clock Depending upon the kinds of information shown in the child s drawing of the clock, ask questions like: Tell me about your clock. What are clocks for? What are the numbers on your clock? (If relevant.) How do the numbers work? What time does your clock show? Tell me what you do at this time. 39 Telling the Time Tell me what time these clocks show? (Use the yellow clock face.) [at any stage] Q40 a) 2:00 b) 9:30 c) 2:20 Interview booklet 39

Mathematics Online Interview 3 40 The Days and the Months a) Tell me the days of the week (or some days if the child hesitates). b) Tell me the months of the year (or some months if the child hesitates). for (a) or (b) Section F c) Tell me what day comes before Friday. F d) Tell me what month comes before April. F Only continue if there is complete success in Q39 and Q40 Otherwise: Section F 41 Calendar Tasks Show the child the yellow calendar card for June. [at any stage] F a) Find the 18th of June. b) Tell me what day of the week that is. c) Show me the last day in June. d) Tell me what month comes after June. e) What day of the week will the first of July be? 42 Duration Tasks Show the child the green page with the diagram of the digital clock showing 12:51. a) You put a pizza in the oven when the clock shows 12:51. You take the pizza out after 13 minutes. What is the time then? If the child answers 13:04, ask the next question. b) Is there another name for that time? Not saying 1:04 Section F 40 Interview booklet