Mathematics Assessment by Group 2

Assessment in Mathematics GROUP 2 Lee Wei, June Lenny Azlina Ong Fei Min, Flora Deborah Tan Yap Thiam Chuan

Overview of Presentation 1. Deciding on a developmental continuum 1.1 The Context, the task & Purpose of task 1.2 Framework of Mathematics in National Curriculum 1.3 Components in National Curriculum framework 1.4 Learning Framework: the Dreyfus model 2. Developing a standards referenced assessment framework 2.1 Building a pedagogical framework 2.2 Identifying performance indicators 2.3 Breaking down the domain 2.4 Quality criteria & initial rubrics 3. Drafting the assessment items 3.1 Questions on test 4. Panelling 4.1 Procedure 4.2 Issues, concerns and themes 4.3 Comments from panellists 5. Revision of Assessment framework and test items 5.1 Quality criteria & amended rubrics 5.2 Further amendments & revised rubrics 5.3 Final performance matrix 5.4 Revised test items 6. Implementation of assessment 6.1 Test conditions 6.2 Instructions for administrator of test 7. Analysis of results 7.1 Zone of proximal development 7.2 Guttman chart 7.3 Analysis: Commonalities 7.4 Analysis: Anomalies 8. Reporting 8.1 Scope of assessment and audience 8.2 Reporting for students & parents 8.3 Reporting for teachers & school 9. Discussion 9.1 Intervention 9.2 Reliability & Validity 9.3 Scaling up 10. Group Reflection 10.1 Reflection on the assessment task 10.2 Reflection on how the interpretation was done 10.3 Reflection: last words

1 Deciding on a Developmental Continuum 1.1 The Context, the Task & Purpose of task 1.2 Framework of Mathematics in National Curriculum 1.3 Components in National Curriculum framework 1.4 Learning Framework: the Dreyfus model

1.1 The Context General profile Autonomous, all-girls’ school Upper Secondary 15 year olds, Express stream Science subject combination Class profile Generally weak in Math Girls are very ‘Arts-driven’ (Drama) and not keen on the subject – preference for the Humanities Students are self-motivated and are highly disciplined

1.1 The task Mathematics Syllabus D Secondary 3 Express stream 30 minutes individual written test Indices & Algebraic Manipulations

1.1 Purpose of task Formative assessment This test is conducted to allow students and teacher to check their knowledge and understanding of Indices and Algebraic manipulation From the assessment, students and teacher can strategize on how to improve in the areas they are weak in.

1.2 Framework for Mathematics in National Curriculum

1.3 Components in Mathematics National Curriculum framework Skills Expansion and factorisation of algebraic expressions (both) Recognising and applying special products Concepts Processes Application Reasoning, communication and connections Thinking skills and heuristics Attitudes Metacognition These three components will be explored in this pilot study

1.4 Learning framework: the Dreyfus Model Novice Rigid adherence to taught rules or plans; Little situational perception; No discretionary judgment Having knowledge and remembering (recalling) of appropriate, previously learned information such as terminology, specific facts or ways and means of dealing with specific conventions and sequences. In this case, having knowledge of the mathematical nomenclatures in standard forms and numbers, and performing simple sequences in manipulation of numbers. Advanced Beginner Guidelines for action are based on attributes or aspects (aspects are global characteristics of situations that can be recognised only after some prior experience); Unable to see the entirety of a new situation (may miss some critical details) Competent Perceives actions at least partially in terms of longer-term goals; Conscious deliberate planning; Standardised and routinised procedures. Uses previously learned information in new and concrete situations to solve problems that have single or best answers. Proficient Sees situations holistically rather than in terms of aspects; See what is most important in a situation; Perceives deviations from the normal pattern and possesses a web or perspectives on a situation. For example when the students are able to identify strategies, differentiate and apply multiple laws of indices. Decision making is less laboured; Uses maxims for guidance, whose meaning varies according to the situation. A maxim is a brief expression of a simple truth, a code of conduct or cryptic instructions that make sense only if there is already a deep understanding of the situation. Expert No longer relies on rules, guidelines or maxims; Operates from a deep understanding of the total situation; Analytical approaches are only used in novel situations or when a problem occurs. The student is able to manipulate complex algebraic expressions combined with laws of indices, and has a vision of what is possible.

2 Developing a standards referenced assessment framework

2.1 Building a Pedagogical Framework Pedagogy DOMAIN: These are sets of skills, knowledge, behaviours and dispositions that enable us to sample and define phases, the strands or constructs STRANDS & CAPABILITY: The big ideas that are learnable, teachable. They too are the set of skills, knowledge and expectations or learning outcomes 1 INDICATOR: Identified behavioural indicators. These indicative behaviours were things that a student could do, say, make or write , and from which we infer their capability in an area CRITERIA: Finally, observational statements that detailed ‘how well’ each behaviour could be performed were created. These criteria underpin the profiling of learning pathways. The criteria are represented by ordered levels of increasing difficulty, sophistication, elegance, etc… 2 3 n

2.2 Performance Indicators Manipulation of numbers in standard form notations Knowledge of mathematical nomenclature in standard form and numbers Comparison of numbers Manipulation of algebraic expressions Application of laws of indices and concepts Mathematical reasoning and communication *Relying on professional knowledge and experience of certain members of the group

2.4 Quality Criteria & Initial Rubrics Manipulation of standard form Manipulation of algebraic expressions Application of Concepts Mathematical reasoning, communication and connections Expert Compare and contrast alternative methods in the use of different laws Explain errors / misconceptions in the: – algebraic manipulation, - use of the laws of indices Proficient Perform the four operations in standard forms expressions Manipulate algebraic fractions and polynomials (with positive, negative, zero and fractional indices), showing consistent and clear working Apply multiple laws of indices concurrently Competent Identify large and small numbers such as giga, micro, pico Manipulate algebraic expressions with positive, negative, zero or fractional indices Differentiate the laws of indices and apply laws appropriately Identify errors in the four operations on indices, surds and use of laws of indices Novice & Advanced Beginner Express numbers in standard form Perform four operations on simple algebraic fractions

2.4 Quality Criteria & Initial Rubrics Students are introduced to the nomenclature, and learn to express familiar numbers in simple standard forms. At competent level, the manipulation of very small and very large numbers would be done. Moving to the proficient level, the students would be able to perform the operations, namely the addition, subtraction, multiplication and division of standard form expressions. At the competent level, students would be able to manipulate algebraic expressions and fractions with indices (or powers). The nature of the math curriculum is spiral, such that students would be taught algebraic manipulation with increasing complexity. At this secondary three level, they would be taught how to manipulate algebraic fractions with indices. The laws of indices is a major concept at upper secondary, so it was not considered as a novice or advanced beginner level. At the competent level, students are expected to be able to differentiate which laws to use, and to apply the laws appropriately. They would only be required to apply the isolated laws. As mentioned in the learning framework, there would be conscious deliberate planning. Students would show standardized and routinized procedures in applying each law. The ability to articulate reasons, and communicate strategies and connections was beyond a novice or advanced beginner level, as students at lower secondary level need only solve questions in routinized manner, and rarely requiring them to state the reasons or laws used. At competent level, the students are able to identify and explain errors in the use of indices. It would be inferred that they have understood and concept, and could communicate their understanding using mathematical reasoning. Manipulation of standard form Manipulation of algebraic expressions Application of Concepts Mathematical reasoning, communication and connections Expert Compare and contrast alternative methods in the use of different laws Explain errors / misconceptions in the: – algebraic manipulation, - use of the laws of indices Proficient Perform the four operations in standard forms expressions Manipulate algebraic fractions and polynomials (with positive, negative, zero and fractional indices), showing consistent and clear working Apply multiple laws of indices concurrently Competent Identify large and small numbers such as giga, micro, pico Manipulate algebraic expressions with positive, negative, zero or fractional indices Differentiate the laws of indices and apply laws appropriately Identify errors in the four operations on indices, surds and use of laws of indices Novice & Advanced Beginner Express numbers in standard form Perform four operations on simple algebraic fractions

3 Drafting the Assessment Items

3.1 Questions on test Question 1: Manipulation of numbers in standard form notations - Comparison of numbers Question 2: Manipulation of numbers in standard form notations - Knowledge of mathematical nomenclature in standard form and numbers Question 3: Manipulation of numbers in standard form notations Question 4 & 5: Manipulation of algebraic expressions Application of laws of indices and concepts Question 6 & 7: Manipulation of algebraic expressions Application of laws of indices and concepts Mathematical reasoning and communication

4.1 Procedure Panelling refers to: checking of the test items and initial rubric using a group (panel) of specialists chaired by one member of the group appointed as a leader Five teachers checked the test items and rubrics. Reviewing / panelling of draft test items: Making specific ideas for change Checking the learning area and year level Identifying content range and gaps Each panel member reviews their item and makes notes about faults and recommendations to fix the fault

Step 2: Panelling 4.2 Issues, concerns and themes Refine questions to target students’ level and syllabus Rephrase questions to avoid ambiguity and biases Adjust criteria and levels

Step 2: Panelling 4.3 Comments from panellists Refine questions to target students’ level and syllabus “ I personally feel that for half hour we should keep to max 6 questions” “ Are the questions too difficult for the students? Are they able to explain errors? Does the syllabus require them to do it?” Rephrase questions to avoid ambiguity or biases “ For question 1, should it be 'rank the following expressions numbers 1 to 4... with 1 representing highest value and 4 lowest " instead of 1 to 4?” “ Not sure which skill / concept question 1 is testing on the rubric - but I'm wondering if there is a chance of a student getting it right by guesswork? Or if they did get it wrong, can we tell the misconception from their answers? i.e. whether they had 1,3,2,4, or 3,2,1,4 etc. If so, should this be reflected in the rubrics?”

Step 2: Panelling 4.3 Comments from panellists (con’d) Adjust criteria and levels, ensure clarity and common understanding in criteria “ I'm wondering for the last column if we can bring up the identifying error from competent to proficient…Also for standard form, there‘re no questions testing this criteria on identifying error…” “ the misconceptions in question 7 don't only specify laws of indices so I'm wondering if this will mean that they first need to have the ability to identify that the error is in the use of the laws?” “ what happens if students explain misconception in question 7, but did not show correct steps? … we will have to determine if the incorrect steps were due to careless mistakes, if it is conceptual, then the student does not achieve proficient level.” “ if student could identify errors in 4 operations (Competent), wouldn’t it mean that he could perform 4 operations in std form (Proficient)?” “ For 'Application of Concepts' column, is it possible for a student to be able to compare and contrast alternative methods (Expert) but not able to apply multiple laws of indices concurrently (Proficient)?” “ For 'Manipulation of Expressions' column, does Competent need to be able to show consistent and clear working as well (compared to Proficient)”

5 Revision of Assessment Framework and Test Items

5.1 Quality criteria & amended rubrics Amended rubric with Katie’s comments on 4 Feb 2010 Indicators > Levels  Manipulation of numbers in standard form notations Manipulation of algebraic expressions Application of Concepts Mathematical reasoning, communication and connections Expert Compares and contrasts alternative methods in the use of different laws, and multiple laws Explains errors / misconceptions in the: – algebraic manipulation, - use of the laws of indices Proficient Performs operations in standard forms expressions Manipulates algebraic fractions and polynomials (with positive, negative, zero and fractional indices), showing consistent and clear working Applies multiple laws of indices concurrently Identify errors in the four operations on indices, surds and use of laws of indices Competent Identifies large and small numbers such as giga, micro, pico Manipulates algebraic expressions with positive, negative, zero or fractional indices, showing clear working. Differentiates the laws of indices and apply laws Novice & Advanced Beginner Expresses numbers in standard form Performs operations on simple algebraic fractions

5.2 Further amendments and revised rubrics Seven individual rubrics were then set up for each question, to allow more clarity in the assessment of the students’ level. At the same time, the questions, as well as the criteria and indicators required further refinement to the indicators and criteria - to reflect the skills and observable behaviours more accurately . From the developmental perspective, we became more mindful of the considerations to be made with regards to the: Content: Can the content (test items) define students’ learning levels? Developmental levels: Can the order of levels reflect the developmental sequence?

5.3 Final performance matrix See slide notes for explanation Manipulation of numbers in standard form notations Knowledge of mathematical nomenclature in standard form and numbers Comparison of numbers Manipulation of algebraic expressions Application of laws of indices and concepts Mathematical reasoning and communication in algebraic manipulation Mathematical reasoning and communication in indices Cut off points Expert Manipulates algebraic fractions and polynomials (with positive, negative, zero indices), showing consistent and clear working Explains strategies used in the algebraic manipulation 13 to 14 Proficient Applies multiple laws of indices concurrently States the types of algebraic manipulation States the operations on indices, surds or laws of indices used in the working 10 to 12 Competent Performs operations involving standard form expressions Manipulates algebraic expressions with positive and negative indices, showing clear working. Differentiates the laws of indices and applies the law(s) 7 to 9 Advanced beginner Manipulate numbers with negative, positive powers and standard forms by moving decimal places Compares small and large numbers by using strategies e.g. converting to common powers or common forms Performs operations on single algebraic expressions 4 to 6 Novice Manipulates numbers with either negative or positive powers by moving decimal place in a single direction Expresses small and large numbers using standard form, giga, micro, pico etc. Compares numbers without showing clear working. 1 to 3

Question 1: Manipulation of numbers in standard form notations Comparison of numbers Question 2: Manipulation of numbers in standard form notations Knowledge of mathematical nomenclature in standard form and numbers Question 3: Manipulation of numbers in standard form notations Question 4: Manipulation of algebraic expressions Application of laws of indices and concepts Mathematical reasoning and communication – in (i) algebra and (ii) indices 5.4 Revised test items

5.4 Revised test items Q1. Rank the following expressions using numbers (1), (2), (3) and (4) in ascending order, with (1) representing the smallest value and (4) representing the largest value. Ranking: ( ) ( ) ( ) ( ) 340 × 10 −4 , 0.000034 × 10 4 , 0.034 × 10 2 , 3·4 × 10 −3 Show Working: Q2. Convert the following lengths to metre, leaving your answers in standard form 240 gigametres = 1370 picometres = Q3. Given that a = 80 × 10 6 , and b =1.9 × 10 4 , express a + b in standard form. Indicator 1: Manipulation of numbers in standard form notations will be shown through students’ working. Indicator 3: Comparison of numbers. At advanced beginner level, students will show strategies via their manipulation. Indicator 1: Manipulation of numbers in standard form notations Indicator 2: Knowledge of mathematical nomenclature in standard form and numbers Indicator 1: Manipulation of numbers in standard form notations. Students at competent level would be able to perform the operation in standard form.

5.4 Revised test items Q4 . Simplify the following algebraic expression, leaving your answers in a single fraction and positive indices. State mathematical reasons and strategies for every step that you take. Write these reasons and strategies on the same line as your working, stating the laws of indices where applicable. Working Reasons/ strategies Indicator 4: Manipulation of algebraic expressions Indicators 6 and 7: Mathematical reasoning and communication – in (i) algebraic manipulation (indicator 6) and (ii) indices (indicator 7) Indicator 5: Application of laws of indices and concepts

6 Implementation of Assessment

6.1 Test conditions Conducted by a trainee teacher During timetabled Mathematics period Students did not have any remedial or re-cap of the topics prior to test as the topic of Indices was newly covered earlier in the year. Algebraic manipulations topic was covered in 2009.

6.2 Instructions for administrator of test All bags & notes are placed in front of the class. Write time duration, start and end on the board. Remind students to do their work individually without reference to any notes. Students should only have their stationery on the table. No calculators are allowed. Distribute the test face-down and instruct students not to turn over until told to do so. Cue for start following class clock, 10 minutes before end and end time. NOTE: the school has a protocol for written tests and exams and all staff will be able to conduct the test in the manner expected.

7.1 Zone of Proximal Development (ZPD) Vygotsky “ ...a state of readiness in which a student will be able to make certain kinds of conceptual connections, but not others; anything too simple for the student will quickly become boring; anything too difficult will quickly become demoralising”. So, when is a student ready to learn? A zone in which an individual can learn more with assistance than he or she can manage alone.

7.2 Guttman Chart At this level, the recommended next intervention is to strengthen manipulation of algebra At this level, the recommended next intervention is to strengthen manipulation of numbers in standard form At this level, the recommended next intervention is to strengthen reasoning and application of multiple laws for indices At this level, the recommended next intervention is to strengthen reasoning for algebra and manipulation of fractional algebra Name 1a 3a 3b 1b 1c 4a 2a 4b 5a 6a 7a 6b 5b 4c Score Level Norazah 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14 5 Ke Tian 1 1 1 1 1 1 1 1 1 1 0 1 1 0 12 4 Marissa 1 1 1 1 1 1 1 1 1 1 1 1 0 0 12 4 Stacy 1 1 1 1 1 1 0 1 0 1 1 1 0 0 10 4 Isabelle 1 1 1 1 1 1 0 1 1 0 1 0 1 0 10 4 Suzanne 1 1 1 1 1 1 0 1 1 0 0 0 1 0 9 3 Kelly 1 1 1 1 1 1 1 1 1 0 0 0 0 0 9 3 Dianne 1 1 1 1 1 1 1 1 0 0 0 0 0 0 8 3 Cherlyn 1 1 1 1 1 1 0 1 1 0 0 0 0 0 8 3 Nasirah 1 1 1 1 0 1 0 1 1 0 0 0 0 0 7 3 Nur Zahwah 1 1 1 1 1 0 0 0 1 0 0 0 0 0 6 2 Eunice 1 1 1 1 1 0 1 0 0 0 0 0 0 0 6 2 Namrata 1 1 1 1 1 0 1 0 0 0 0 0 0 0 6 2 Amanda 1 1 1 1 1 0 0 0 0 1 0 0 0 0 6 2 Celestine 1 1 1 1 0 1 0 0 0 0 0 0 0 0 5 2 Sharifah 1 1 1 1 0 0 1 0 0 0 0 0 0 0 5 2 Peo Shan 1 1 1 1 1 0 0 0 0 0 0 0 0 0 5 2 Cheryl 1 1 1 1 1 0 0 0 0 0 0 0 0 0 5 2 Nicole 1 1 1 1 0 0 0 0 0 0 0 0 0 0 4 2 B. Rohini 1 1 1 1 0 0 0 0 0 0 0 0 0 0 4 2 Gitanjali 1 1 1 1 0 0 0 0 0 0 0 0 0 0 4 2 Annetta 1 1 1 0 0 0 0 0 0 0 0 0 0 0 3 1 41 41 40 37 24 23 19 18 15 13 10 10 8 6

7.2 Guttman Chart Name 1a 3a 3b 1b 1c 4a 2a 4b 5a 6a 7a 6b 5b 4c Score Level Norazah 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14 5 Ke Tian 1 1 1 1 1 1 1 1 1 1 0 1 1 0 12 4 Marissa 1 1 1 1 1 1 1 1 1 1 1 1 0 0 12 4 Stacy 1 1 1 1 1 1 0 1 0 1 1 1 0 0 10 4 Isabelle 1 1 1 1 1 1 0 1 1 0 1 0 1 0 10 4 Suzanne 1 1 1 1 1 1 0 1 1 0 0 0 1 0 9 3 Kelly 1 1 1 1 1 1 1 1 1 0 0 0 0 0 9 3 Dianne 1 1 1 1 1 1 1 1 0 0 0 0 0 0 8 3 Cherlyn 1 1 1 1 1 1 0 1 1 0 0 0 0 0 8 3 Nasirah 1 1 1 1 0 1 0 1 1 0 0 0 0 0 7 3 Nur Zahwah 1 1 1 1 1 0 0 0 1 0 0 0 0 0 6 2 Eunice 1 1 1 1 1 0 1 0 0 0 0 0 0 0 6 2 Namrata 1 1 1 1 1 0 1 0 0 0 0 0 0 0 6 2 Amanda 1 1 1 1 1 0 0 0 0 1 0 0 0 0 6 2 Celestine 1 1 1 1 0 1 0 0 0 0 0 0 0 0 5 2 Sharifah 1 1 1 1 0 0 1 0 0 0 0 0 0 0 5 2 Peo Shan 1 1 1 1 1 0 0 0 0 0 0 0 0 0 5 2 Cheryl 1 1 1 1 1 0 0 0 0 0 0 0 0 0 5 2 Nicole 1 1 1 1 0 0 0 0 0 0 0 0 0 0 4 2 B. Rohini 1 1 1 1 0 0 0 0 0 0 0 0 0 0 4 2 Gitanjali 1 1 1 1 0 0 0 0 0 0 0 0 0 0 4 2 Annetta 1 1 1 0 0 0 0 0 0 0 0 0 0 0 3 1 41 41 40 37 24 23 19 18 15 13 10 10 8 6

Guttman Chart Name 1a 3a 3b 1b 1c 4a 2a 4b 5a 6a 7a 6b 5b 4c Score Level Norazah 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14 5 Ke Tian 1 1 1 1 1 1 1 1 1 1 0 1 1 0 12 4 Marissa 1 1 1 1 1 1 1 1 1 1 1 1 0 0 12 4 Stacy 1 1 1 1 1 1 0 1 0 1 1 1 0 0 10 4 Isabelle 1 1 1 1 1 1 0 1 1 0 1 0 1 0 10 4 Suzanne 1 1 1 1 1 1 0 1 1 0 0 0 1 0 9 3 Kelly 1 1 1 1 1 1 1 1 1 0 0 0 0 0 9 3 Dianne 1 1 1 1 1 1 1 1 0 0 0 0 0 0 8 3 Cherlyn 1 1 1 1 1 1 0 1 1 0 0 0 0 0 8 3 Nasirah 1 1 1 1 0 1 0 1 1 0 0 0 0 0 7 3 Nur Zahwah 1 1 1 1 1 0 0 0 1 0 0 0 0 0 6 2 Eunice 1 1 1 1 1 0 1 0 0 0 0 0 0 0 6 2 Namrata 1 1 1 1 1 0 1 0 0 0 0 0 0 0 6 2 Amanda 1 1 1 1 1 0 0 0 0 1 0 0 0 0 6 2 Celestine 1 1 1 1 0 1 0 0 0 0 0 0 0 0 5 2 Sharifah 1 1 1 1 0 0 1 0 0 0 0 0 0 0 5 2 Peo Shan 1 1 1 1 1 0 0 0 0 0 0 0 0 0 5 2 Cheryl 1 1 1 1 1 0 0 0 0 0 0 0 0 0 5 2 Nicole 1 1 1 1 0 0 0 0 0 0 0 0 0 0 4 2 B. Rohini 1 1 1 1 0 0 0 0 0 0 0 0 0 0 4 2 Gitanjali 1 1 1 1 0 0 0 0 0 0 0 0 0 0 4 2 Annetta 1 1 1 0 0 0 0 0 0 0 0 0 0 0 3 1 41 41 40 37 24 23 19 18 15 13 10 10 8 6

7.3 Analysis: commonalities Pupils Learning difficulties Possible interventions This applies to all students except Norazah. For example, both Isabella & Suzanne could apply multiple laws of indices concurrently, but seemed to have difficulty in reasoning for algebra in this assessment. General observation of greater difficulty in algebra compared to indices. Revision of concepts and understanding 14 out of 22 students These students seemed to have difficulty with the language for Math i.e. Identifying large and small numbers such as giga, pico. Emphasis of the importance to remember the language

7.4 Analysis: anomalies Pupils Learning difficulties Possible interventions Ke Tian She seemed to have problems stating the laws of indices in this assessment, though she was able to apply multiple laws. Cause – Problem with language Intervention – Enhancing the use of math language Stacy She seemed to have problems differentiating the laws of indices in this assessment, though she was able to apply multiple laws. Cause – Learning the steps by rote learning Intervention – Starting from the laws of indices, students learn how questions can be derived from individual laws. Amanda She seemed to exhibit reasoning for strategies she used. She had problems with manipulation and application of indices and algebra in this assessment. Cause – Understanding of the concepts were not in depth Intervention – Enhancing conceptual understanding Nur Zahwah She seemed to be able to differentiate the laws of indices. She had problems with manipulation and application of indices and algebra in this assessment. Cause – Understanding of the concepts were not in depth Intervention – Enhancing conceptual understanding

8 Reporting 8.1 Reporting for students and parents 8.2 Reporting for Math teacher and school

Reporting Since this assessment is formative in nature, the results of the assessment would be helpful to the individual students to help them develop their competencies. There are two parts to the individual report: Rocket report The individual report on the performance on each indicator For the teacher another report with the performance of students grouped by competencies would be generated too.

8.1 Scope of Assessment & Audience Scope of Assessment Term Tests & Semester Examinations Class Tests Students & Parents Teachers School

Reporting audience Section Method of reporting Purpose / Implications Subject teacher 8b Student performance by indicators Student overall performance descriptors Pick out anomalies and commonalities Categorise whole class intervention, remediation, buddy or individual Report to other teachers and/or HOD HOD Maths / Buzz team (school) Student performance by indicators Student overall performance descriptors Adjust SOW to include past year revision which serves as foundation for current topics Review teachers’ teaching and assessment abilities for testing the concepts, skills, processes aligned to national framework.* Parents 8a Student rocket report which includes intervention suggestion Monitor that student does the work at home and support the intervention Student Student rocket report Student individual performance Student will be able to strategise on the area they need to work on. Check their own progress.

8.1 Reporting for students and parents

Performance Indicators Student individual progress report: Norazah (28) Manipulation of numbers in standard form notations Knowledge of mathematical nomenclature in standard form and numbers Comparison of numbers Manipulation of algebraic expressions Application of laws of indices and concepts Mathematical reasoning and communication in algebraic manipulation Mathematical reasoning and communication in indices Suggested intervention: Well done! Do work on your Mathematical metacognitive abilities by verbalising strategies with your partner. Parent’s signature: Date: ___________ A B Class performance middle 50% Norazah Level 5 B. Student can perform simple sequences in the manipulation of numbers (i.e. with either positive or negative powers). D. Student can manipulate numbers in standard form and algebraic expressions. Student is able to routinely apply the law(s) of indices. F. Student can perform and explain the manipulation of complex algebraic expressions combined with laws of indices . Student is proficient in demonstrating clear strategies and stating their analytical approaches. that deep understanding of application of the laws of indices. C. Student can perform manipulation of numbers and single algebraic expressions. Student is able to compare magnitude of numbers using clear strategy. E. Student can manipulate numbers in standard form and algebraic expressions. Student is proficient in the application of multiple laws of indices concurrently . A. There is no evidence of student’s ability to manipulate numbers and algebraic expressions. C D E F

Performance Indicators Student individual progress report: Celestine (6) A B Class performance middle 50% Celestine Level 2 B. Student can perform simple sequences in the manipulation of numbers (i.e. with either positive or negative powers). D. Student can manipulate numbers in standard form and algebraic expressions. Student is able to routinely apply the law(s) of indices. F. Student can perform and explain the manipulation of complex algebraic expressions combined with laws of indices . Student is proficient in demonstrating clear strategies and stating their analytical approaches. that deep understanding of application of the laws of indices. C. Student can perform manipulation of numbers and single algebraic expressions. Student is able to compare magnitude of numbers using clear strategy. E. Student can manipulate numbers in standard form and algebraic expressions. Student is proficient in the application of multiple laws of indices concurrently . A. There is no evidence of student’s ability to manipulate numbers and algebraic expressions. Manipulation of numbers in standard form notations Knowledge of mathematical nomenclature in standard form and numbers Comparison of numbers Manipulation of algebraic expressions Application of laws of indices and concepts Mathematical reasoning and communication in algebraic manipulation Mathematical reasoning and communication in indices Suggested intervention: Work with your partner to work on the performance indicators checklist. Parent’s signature: Date: _______________ C D E F

Summary statements of learning at each level Level 4: At this level, the student is learning to apply multiple laws of indices concurrently. She/he is also learning to state the types of algebraic manipulation and the operations on indices, surds or laws of indices used in the working. Level 1: At this level the student is learning to manipulate numbers with either negative or positive powers by moving decimal place in a single direction. She/he is learning to use the knowledge of common terms like standard form, small and large numbers such as giga, micro, pico. She/he is also learning to compare numbers without showing clear working. Level 5: At this level, the student is learning to manipulate algebraic fractions and polynomials (with positive, negative, zero indices), showing consistent and clear working. She/he is also learning to explain strategies used in the algebraic manipulation. Level 3: At this level, the student is learning to manipulate algebraic expressions with positive, negative, zero or fractional indices, showing clear working. She/he is also learning to differentiate the laws of indices and applying laws. . Level 2: At this level the student is learning to manipulate numbers with negative and positive powers by moving decimal places. She/he is also learning to compare numbers by using strategies e.g. converting to common powers or common forms. She/he is also learning to perform operations on simple algebraic fractions. Category: aesthetics Performance level descriptors Expert 13-14 Proficient 10-12 Competent 7-9 Advanced Beginner 4-6 Novice 1-3 Indicators  1. Manipulation of numbers in standard form notations 2. Language of common terms in standard form and number 3. Comparison 4. Manipulation of algebraic expressions 5. Application of concepts 6. Mathematical reasoning, communication for algebra 7. Mathematical reasoning, communication for indices

Student rocket report A B C D E F 50% of the class can be located within this range The student is estimated to be at the location Inter-quartile range Student Achievement Level Level descriptions Levels B. Student can perform simple sequences in the manipulation of numbers (i.e. with either positive or negative powers). D. Student can manipulate numbers in standard form and algebraic expressions. Student is able to routinely apply the law(s) of indices. F. Student can perform and explain the manipulation of complex algebraic expressions combined with laws of indices . Student is proficient in demonstrating clear strategies and stating their analytical approaches. that deep understanding of application of the laws of indices. C. Student can perform manipulation of numbers and single algebraic expressions. Student is able to compare magnitude of numbers using clear strategy. E. Student can manipulate numbers in standard form and algebraic expressions. Student is proficient in the application of multiple laws of indices concurrently . A. There is no evidence of student’s ability to manipulate numbers and algebraic expressions.

28. NORAZAH Performance Indicators

6. CELESTINE TAN Performance Indicators

8.2 Reporting for teachers and school Student performance by indicators : teachers can sieve out how to help students progress in specific areas

Student performance by indicators

9 Discussion 9.1 Intervention 9.2 Reliability& Validity 9.3 Scaling up

When we talk about general interventions, what do we need to consider? The students’ individual general developmental levels The students’ group developmental levels The assessment history of the students. Student Results At this level the recommended intervention for the group is... At this level, the recommended intervention for this student is… Developmental level = x . Group Targets Individual targets

Target students Intervention strategies Area owner / description Resources Whole class then one-to-one Review results Teacher- student individual conferencing Teacher to review results and compare to previous results as well as prior knowledge about students to sieve out ‘abnormal’ results. Call students individually to ask what they were going through when they did the paper. Establish if the error is based on mathematical misconception. Teacher training to analyse based on individual profiles. Whole class Include revision before teaching new proper HOD to review input of previous year revision prior to new topic in SOW (teacher to recommend). Review inclusion of metacognition and building mathematical language during lessons. Consider including certain types of questions in formative assessment. Teacher training in building mathematical language teaching and assessment. Sharing during buzz sessions. Teachers need to have a database of questions they can tap on to be used for other intervention strategies. From the maths sharing portal, pick out questions that test different skills, concepts, processes etc and categorise these questions. Whole class n>20 Review of topic Teacher to take one period to go through the important skills or go through lesson to build mathematical language. Affected students 10<n<20 Remediation based on indicators Teacher to go through similar questions and common misconceptions Affected students 1<n<10 Short term buddy system based on indicators Individual students would be paired up with students who have been identified to be able to do these sub-sections well to verbalise strategies and try similar questions together at own time. Student need to verbalise strategies to buddy. Checklist / rubrics for buddy and partner to work on so they can monitor their own progress and report to teacher during stipulated times. Level 1 students One-to-one attention with teacher: face to face & online Students to meet up with teacher to go through the questions they have problems in. Have online questions to do individually. Teacher can monitor students’ progress online. Online learning portal* All students have access to computer and internet.

Validity 9 Type of validity Suggestions for improvement Content validity - This assessment was showed that the content of the assessment tasks were closely related to the school syllabus and subject matter (national math syllabus). This was also contributed by the rich teaching experiences of three math teachers, including the assessor. Use of multiple tasks and multiple sources of evidence as the basis for judgment.

Validity: alignment to national syllabus

Validity: alignment to school syllabus 2010 Secondary 3 Specific Instructional Objectives New Express Mathematics Suggested Activities (Including Teaching Processes) RESOURCES [PD] STRATEGIC FOCUS (Aesthetic, NE, HOM, ICT) Week Learning Outcomes Exercises Exploration Thinking Writing IT/HOM NE/Aesthetic 1-4 (6 periods) Chapter 1: Indices Understand and use concept of indices and laws of indices Understand and use concept of indices and laws of indices Manipulate zero and negative indices Manipulate fractional indices Rewrite and express numbers in standard form Express very large or small numbers Solve problems involving indices Ex 1.1: 4, 5, 6, 7, 8 Ex 1.2: 1, 2, 3, 4, 5, 7, 9 Ex 1.3: 1, 2, 5, 7, 8 Ex 1.4: 3, 4, 5, 6 Ex 1.5: 2, 3, 4, 5 Prefixes and Powers of 10 (pg 24) Ex 1.6: 1 Chapter Review: 6, 11, 12 , 13 In  class Activity pg 8, 11 In  class Activity pg 19 In  class Activity pg 22 Exploration Task pg 29 [Curriculum Differentiation] Thinking Time pg 6 Thinking Time Pg 11 Thinking Time pg 15, 16 Thinking Time pg 23 NE [1] Use standard form to represent big numbers and tie in with size of Singapore population – impact on space, economy Ref: Pg 23 Example 3, pg 25 #5, #6 4-6 (6 periods) Chapter 2: Solving Quadratic Equations Solving quadratic equations by factorization (revision of Sec 2) Solve quadratic equations by completing the square Understand the quadratic formula and use it to solve quadratic equations Solve fractional equations that can be reduced to quadratic equations Solve problems involving quadratic equations Ex 2.1: 3, 4, 7 Ex 2.2: 3, 4, 5 Ex 2.3: 1, 3 Ex 2.3: 2 Ex 2.4: 6 , 8 , 10 Chapter Review: 10, 11 Exploration Task pg 49 [Cooperative Learning] Looking Back pg 32 Thinking Time pg 42 ICT[1]- Graphmatica

Reliability 9 Type of Reliability Suggestions for improvement The overlap between 0s and 1s is not fairly wide (Guttman chart ) which shows fair amount of consistency. A fairly reliable assessment allows a more clearly defined ZPD. Standard administration – The team established and documented clear assessment procedures/instructions for collecting, analysing and recording outcomes. The team used multiple tasks of evidence as the basis for judgment. Inter-rater reliability through paneling – There was a consistency of judgement and moderation of the judgements across different team members using the same assessment task and procedure. Involvement of expertise - The team members and assessor are experienced teachers, three of them are specialised in teaching math and one curriculum officer who have demonstrated competence in the field. Elimination of noise - ‘Noise’ due to individual bias is eliminated when the team reviewed and moderated the competencies in the assessment framework. The team members reflected on their judgement error in competency based assessment and biases. Reliability could be improved with the help of others. As we believe that professional development is social in nature, team effort is useful in helping to improve the reliability. We could use assessors with expertise in competency based assessment. Maintain representative sample of assessment tasks to compare from context to context/year to year and use a panel of independent assessors to evaluate this sample. Use multiple sources of evidence as the basis for judgment.

Scaling up 9 There are issues to consider when scaling up from class test to school formal assessments or from one departmen t to many or from written tests to projects and presentations. For example, will the leadership support teacher training or create more platforms for teachers to discuss in a professional learning team etc. These complexities will be discussed in details in the following slides.

Scaling Up Scaling up in standards based assessment At school level : Use of quality criteria and Guttman charts for future term tests across levels and across department. Implications on: School policy and leadership Teaching practices Teacher PD and competencies Student learning and engagement Parent support 9

1. School Policy, Structure & Culture School leadership has influence on school policy, which determine the structure and culture that would support the standards based assessment. Structures Provide time for teachers to collaborate, analyze data and student work within Professional Learning Communities / Teams The construction within a school organization such as role and job descriptions as well as decisions on time and space, clarity of the goals and means forms the fundamental essence to the success of this educational initiative (Evans, 1996). The rationale and procedures associated with initiating the change should be constantly conveyed clearly throughout the implementation process (Evans, 1996). School Cultures When schools promote this initiative, they inevitably bring about a significant change in the school culture.

The program will not succeed if the Principal is: Disinterested Aware Interested Supportive The program will succeed if the Principal is: Involved Engaged Committed Ownership Support and understanding from school leadership to create a conducive environment for cultivating trust and collegiality 9 1. School Leadership

As assessment carries a lot of stake in Singapore, it is important that the school leaders are willing to support such scaling up. In addition, with the support of the school leaders, resources such as time would be given for the teachers to work on the assessment. Furthermore, we could then look at possibly scaling up to other subjects. 1. Support of School Leaders

2. Teaching Practices Teaching practices needs to be evidence based . Teachers need to focus on strategies, as well as intervention strategies, for differentiated teaching . Teachers need to model the skills that they wish for students to exhibit, e.g. teachers demonstrating reasoning of process. Teachers need to learn to work collaboratively, sharing resources , ideas and keeping abreast of new changes and research. Other teacher factors include: Teacher beliefs and attitudes on Class organisation Differentiated teaching Targets for all students Teachers’ pedagogical skills in Individualised learning Flexible use of resources Teachers’ knowledge of Discipline expertise Developmental learning Assessment and reporting Use of data Teaching or Pedagogy Practices Evidence Based Intervention Strategy Sharing of resources

3. Teacher PD Equip teachers PD in these 5 areas: Leadership training in leading PLT, facilitating data dialogues to help teachers move beyond what the data mean to actions that will close the gap. Use of data to examine data, understand students’ strengths and weaknesses, and identify interventions. Equip teachers in assessment competency , provide guide and coaches (e.g. master teachers, subject specialists) PLT strategies , to create a non-threatening, supportive environment that encourages and allows teachers to be open and honest in planning, assessment, data analysis and reporting. Knowledge of the developmental theory , which provides a common language and common theoretical framework. Note on PD: Follow-up support to teachers needs to be continuous and on-going Teacher PD in 5 areas PLT Strategies Assessment and Reporting Data Use Leadership Developmental Theory

3. Teacher Competencies As mentioned in PLTs, we could get the teachers to go through the whole process of assessment from designing of rubrics to designing of questions to implementation and analyzing the data to reporting and intervention. The key is about learning and so the number of questions could be kept small.

3. Teacher Competencies When the teachers gained the greater competencies and confidence, we could further scale up by having more rubrics and more questions in the tests and finally incorporate the questions into semester examinations.

4. Student learning & engagement This requires a cultural shift towards formative practices of assessments in four areas: questioning, feedback through marking, peer-and self-assessment (Black & William, 1998b) Going beyond processes, skills and concepts to includes attitudes and metacognition (refer National Curriculum framework; interventions by peer coaching as suggested by the interventions)

4. Scaling up process at school level (an example) 9

5. Support of Parents We also need to engage the parents to allow them to understand and appreciate the way assessment is done. With the support of the parents, intervention would be more effective and this would further enhance the confidence of the parents in the assessment process.

5. Support of Parents For example, parents could provide the following support: Setting up interest groups or Parent Support Group (PSG) Local support group Close communication with the teachers and school Monitor and encourage children at home

9 10.1 Reflection on the assessment task Reflection point Thoughts/Implications Clarity of Questions The way the questions was being asked could confuse the questions and mask the competencies of the students. The way the questions was being asked could also guide the students to work aspects such as meta-cognition. Balance between Space for Alternatives and Specific Skills to be Tested. Questions are to be asked to test the ability of the students in that competency. In this case, the assessors will need to be prepared to accept all other plausible acceptable methods that arrive at correct answer, which would not reflect the students competency in standard form.

10.2 Reflection on how the interpretation was done 9 Reflection point Thoughts/Implications Importance of Teacher Judgment There were many situations where it was not clear cut that the students demonstrated or did not demonstrate the competency. Teacher’s expertise and experience are important to make accurate judgments. This implied that the teacher needs to be an expert and experienced one. If not, having more than one marker would help in both getting the more accurate judgments and developing the competencies of the teacher. Understanding the continuity of development We should not just look at this assessment as an isolated event. Instead, knowing the students and their development in Mathematics as a whole would help the teacher develop better intervention for the students.

Seeing the trees and the forest Bearing in mind the linkages between the objectives, assessment rubrics, reporting, and intervention has helped us gain a perspective of a holistic assessment process. This new perspective not only enables us to see assessment as an integral of teaching but also how it could be done. 9 Reflection: Last words

References Dreyfus, H. L. & S. E. Dreyfus. (2004). From Socrates to Expert Systems: The Limits and Dangers of Calculative Rationality: Regents of the University of California. http://socrates.berkeley.edu/~hdreyfus/html/paper_socrates.html Gillis, S. & Bateman, A. (1998). Assessing in VET: Issues of validity & reliability. Review of Research. Griffin, P. (2006). Strategies for developing measures of the skill levels 1. University of Melbourne. Assessment Research Centre. Black, P., Harrison, C., Lee, C., Marshall, B., & William, D. (2003). Assessment for learning. England: Open University Press.

Mathematics Assessment by Group 2

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