PROVINCIAL ASSESSMENT PROGRAM - New Brunswick · New marking logistics were also piloted in the 2010 marking sessions. In the past, all booklets were circulated randomly amongst all

PROVINCIAL ASSESSMENT PROGRAM Provincial Mathematics Assessment Program: Information Bulletin February, 2011 Assessment and Evaluation Branch Department of Education and Early Childhood Development Province of New Brunswick P.O. Box 6000 Fredericton, N.B. E3B 5H1 Canada www.gnb.ca/education

http://www.gnb.ca/education�

_____________________________________________________________________________________________ 1 NB Department of Education and Early Childhood Development

Assessment and Evaluation Branch February, 2011

Table of Contents Overview

Content ...................................................................... Page 2 Competencies ............................................................. Page 3 Marking and Reporting................................................ Page 3

2010 Assessments

Background................................................................. Page 4 2010 Pilot Information ................................................ Page 5

Grade 3 Pilot ................................................... Page 5 New Constructed-Response Format ................. Page 5

New Mental Mathematics Format ................... Page 6 Marking and Reporting …................................. Page 6 2010 Provincial Results ………………………………….. Page 7

2011 Assessments

Timing ………................................................................. Page 8 Administration ……………............................................... Page 8 Marking ….................................................................. Page 9 Rubrics ………………………………………………………………………. Page 9 Reporting ………………………………………………………………….. Page 11

APPENDICES

A. Exemplars............................................................. Page 15 B. Common Errors (2010) .......................................... Page 28

2 NB Department of Education and Early Childhood Development


Overview Assessment and evaluation is an integral component of the teaching and learning process. Assessment enables teachers to gather data to determine the needs of their students and to address those needs adequately in order to tailor instruction. Large-scale data gathered through the provincial assessment program enables policy makers to make programming decisions at the provincial, district or school level. Historical and longitudinal data provide information that may inform policy, professional development, and/or strategies to improve instructional practice. The provincial mathematics assessment program makes every effort to provide information on the goals of the New Brunswick Mathematics Curriculum (2010), which includes preparing students to:

• use mathematics confidently to solve problems, • communicate and reason mathematically, • appreciate and value mathematics, • make connections between mathematics and its applications, • commit themselves to lifelong learning, • become mathematically literate adults, • use mathematics to contribute to society (p. 3). Content Numeracy assessments are conducted at grade 3 (pilot), grade 5 and grade 8 in both the English Prime and French Immersion programs. The assessments require students to respond independently to selected-response items, and to generate answers to open-response questions. Each question is selected or created by New Brunswick educators and are aligned to the New Brunswick Mathematics Curriculum. The areas for assessment are based upon the General Curriculum Outcomes of the New Brunswick Mathematics Curriculum:

Strand Percentage of Assessment*

Number • Number Sense

40 - 50%

Patterns and Relations • Patterns • Variables and Equations

10 - 15%

Shape and Space • Measurement • 3-D Objects and 2-D Shapes • Transformations

20 - 30%

Statistics and Probability • Data Analysis • Chance and Uncertainty

10 - 15%

* Percentages reflect the range within each of the different grade level curricular outcomes.



Competencies A sound mathematical understanding is central in preparing our students for life in modern society. It is thus important to ensure students are adequately prepared to apply mathematical knowledge and to communicate mathematically when solving problems. To ensure the mathematics assessment program provides information on the mathematical literacy of our students, all questions are aligned to measure competency in knowledge, problem solving and communication. Each of these competencies are defined as follows: Knowledge: The ability to recall factual and procedural knowledge with accuracy. Problem Solving: The ability to extend prior knowledge to understand and solve problems. To make connections between and within the different strands of mathematics (number, patterns and relations, shape and space, statistics and probability) and to employ efficient, methodical and organized thought processes to reach reasonable solutions. Communication: The ability to use precise mathematical language, units, symbolic notation, models and graphic representations to communicate with clarity. Marking and Reporting Open-response items are marked by a committee of teachers selected from across the province. The committee reviews each of the rubrics and establishes consistent standards for marking (see Appendix A). Upon completion of the marking sessions, the booklets are sent for scanning and computer compilation. The results are reported per strand and competency and are reported at the school, district and provincial level.



2010 Assessments The following provides information from the provincial mathematics assessment in grades 3, 5 and 8 from the 2009-2010 school year. 2010 Background Information As a result of changes to the provincial mathematics curriculum, it was necessary to redevelop the mathematics assessment program. As a result of the new curriculum, and based on consultations with stakeholders within the public education system, the following changes were implemented: • a realignment of assessment items to the new curricular outcomes, • an increased emphasis on problem solving and communication as outlined in the process standards

of the curriculum, • the elimination of the timed aspect of the mental math component, • the inclusion of a basic facts section for grades 3 and 5, • the introduction of a mathematics assessment in grade 3, • the introduction of scripted administration, • the introduction of a general rubric for marking constructed response items, • the introduction of new marking systems. The following table outlines the structure and content of the 2010 assessments in the three different assessment years:

Provincial Mathematics Assessment at Grade 3 * June, 2010

Provincial Mathematics Assessment at Grade 5 June, 2010

Provincial Mathematics Assessment at Grade 8 June, 2010

Timed Basic Facts Assessment • One-minute timed assessment • 12 questions (basic facts)

Timed Basic Facts Assessment • One-minute timed assessment • 15 questions (basic facts)

Booklet A: Calculator • 60 minutes • 38 selected-response • 1 constructed-response question

Part A: Selected Response • 60 minutes** • 30 selected-response questions

Part A: Selected Response • 60 minutes • 42 selected-response questions

Mental Mathematics Assessment • 15 minutes • 10 questions requiring students to explain

their mental mathematics strategies (sample attached)

Part B: Constructed Response • 60 minutes • Mental Mathematics • Operations • Problem Solving Samples of the constructed-response items are attached.

Part B: Constructed Response • 60 minutes • Mental Mathematics • Operations • Problem Solving Samples of the constructed-response items are attached.

Booklet B: Non-Calculator • 60 minutes • Non-Calculator selected response • Operations constructed response • Problem Solving constructed response Samples of the constructed-response items are attached.

* In the 2010 assessment cycle, all questions in the grade 3 assessment were vetted through both the Atlantic Canada Mathematics Curriculum and the incoming New Brunswick Mathematics Curriculum (WNCP). **As per Protocols for Exemptions and Accommodations, students were provided extra time for all sections (with the exception of the Basic Facts Assessments in grades 3 and 5); up to 100% of the specified time per component.



5 NB Department of Education and Early Childhood Development Assessment and Evaluation Branch

March, 2011

2010 Pilot Information As outlined in the Standards of Fair Assessment Practices, all new components of the assessments were piloted to ensure the reliability and validity of assessment items. The pilot was run as part of the general assessment; therefore, all students in the Province of New Brunswick completed the piloted items and all schools received the new Teacher Guide outlining procedures for the scripted administration. Of the entire provincial population at each grade level, a representative sample was drawn for analysis. The sample was established to ensure an equal distribution of urban and rural populations, English and French Immersion programming, and school district representation. The purpose of a pilot is to gather information on the performance of the assessment and is used internally to determine reliability of items, administrative procedures and marking systems. Therefore, pilot information is not released publicly. Grade 3 Pilot Through consultation with educational stakeholders, a consensus was reached determining the implementation of an assessment in grade 3 to coincide with the new mathematics curriculum. However, during this development, additional programming changes occurred in grade 3; specifically, the change from Grade 1 French Immersion entry to Grade 3 French Immersion entry. Due to these programming changes, results for the Provincial Mathematics Assessment at Grade 3 will be held for provincial programming information only and will not be released. Being the first year of full implementation, the Provincial Mathematics Assessment at Grade 3 will continue to run as a pilot for the 2010-2011 school year. New Constructed-Response Format According to the New Brunswick Mathematics Curriculum, there are critical components student must encounter in a mathematics program in order to achieve the goals of mathematics education and encourage lifelong learning in mathematics (p. 7). These components are outlined in the Mathematical Processes in the front-matter of the curriculum document and include an expectation that students will:

• communicate in order to learn and express their understanding of mathematics • connect mathematical ideas to other concepts in mathematics, to everyday experiences and to

other disciplines, • demonstrate fluency with mental mathematics and estimation, • develop and apply new mathematical knowledge through problem solving, • develop mathematical reasoning, • select and use technologies as tools for learning and solving problems, • develop visualization skills to assist in processing information, making connections and solving

problems (p. 7). In order to meet these standards, it was necessary to redesign the constructed-response questions within the Provincial Mathematics Assessments. Specifically, the assessments moved towards a new emphasis on problem-solving items that encourage: proportional reasoning; the use of prior learning in new ways and contexts; making connections between the different representations, strands and topics in mathematics; the demonstration of logical, systematic thinking and, intuitive and inductive reasoning



6 NB Department of Education and Early Childhood Development Assessment and Evaluation Branch

March, 2011

based on patterns and regularities; recognizing and extracting the mathematics embedded in the situation (to mathematize); analyzing, interpreting, and developing their own models and strategies; and, making mathematical arguments, including proofs and generalizations (NBMC, 2010; TIMSS, 2007; PISA, 2002). The new question types are designed with these goals in mind and follow a framework which includes an emphasis on cross-curricular, context based questions that require students to synthesize their mathematical knowledge.

Mental Mathematics

During the original consultation process, it was determined that the method used for measuring mental mathematics knowledge on the provincial assessments was not generally accepted across the province; in particular, the timed portion of the assessment. Although, by definition, mental mathematics is a process that does not involve paper and pencil, it was felt that it was necessary to maintain the mental mathematics component on the provincial assessments due to the importance of the process. As a result, a team of numeracy leads, teachers and learning specialists (representing all districts), in consultation with educational measurement specialists, developed a method for measuring mental mathematics knowledge in the provincial assessment format. The method involves students explaining the mental processes they used when solving a problem. The response is then coded using a rubric. When coding a student response to a mental mathematics question, teacher markers identify the strategy and evaluate the success of the method used. This method of measuring mental mathematics was piloted in all assessments in 2010; consequently, the results for these questions were not reported in 2010. 2010 Marking Sessions

The purpose of the 2010 summer marking sessions was to test the new question types and rubric. Of the entire provincial population at each grade level, a representative sample of student booklets was drawn for analysis. The sample was established ensuring an equal distribution of urban and rural populations, English and French Immersion programming, and school district representation. As a result of the sampling, the summer marking sessions consisted of a reduced number of markers. New marking logistics were also piloted in the 2010 marking sessions. In the past, all booklets were circulated randomly amongst all teachers independently marking. The new logistics involve markers working in small table groups. Within this new system, reliability is ensured through small group and pair marking. 2010 Reporting

As mentioned, due to the changes in curriculum, items, item format, and marking, it was necessary to run pilots for the new components (constructed-response items) of the 2009-2010 assessments. However, to maintain longitudinal trend information for grades 5 and 8, a statistically significant portion of the multiple-choice questions were maintained (all multiple-choice items were aligned to the new curriculum). All other multiple-choice items were analyzed for reliability and the entire multiple-choice section (with the exception of a few non-performing items) was used in the 2010 provincial reporting. Grade 5 and grade 8 results (based on selected-response items only) were released to schools in October. Longitudinal strand information was not reported in 2010 due to the change in curriculum and subsequent changes to the strand structure.



2010 Provincial Results by Grade Level

Further information regarding Provincial Achievement results can be found on the Department of Education and Early Childhood Development website at: http://www.gnb.ca/0000/results/index_e.html

34.837.5

27.7

40.637.1

22.3

40.737.3

22

0

10

20

30

40

50

Perc

enta

ge

Provincial Mathematics Assessment at Grade 5 Achievement Levels

2008

2009

2010

65.259.4 59.3

30405060708090

100

2008 2009 2010

Perc

enta

ge

Provincial Mathematics Assessment at Grade 5Percentage of Students at Appropriate or Above

Below Appropriate Achievement

Appropriate Achievement

Strong Achievement

43

35.7

21.3

41.437.8

20.8

37.4 37.6

21.7

01020304050

BA AA SA

Perc

enta

ge

Provincial Mathematics Assessment at Grade 8Achievement Levels

2008

2009

2010

57 58.6 59.3

30405060708090

100

2008 2009 2010

Perc

enta

ge

Provincial Mathematics Assessment at Grade 8Percentage of Students at Appropriate or Above

Below Appropriate Achievement

Appropriate Achievement

Strong Achievement

66.168.7 68

6560 59.167.2 67.3 66.5 64.9

58.9 59.550556065707580

04-05

05-06

06-07

07-08

08-09

09-10

% A

ppro

pria

te a

nd A

bove

Provincial Mathematics Assessment at Grade 5Perfomance by Gender - Historical

f

m

57.855.4 57.1 55.7 57.2 59.1

64.160.1 59 58.2 60 59.4

50556065707580

04-05

05-06

06-07

07-08

08-09

09-10

% A

ppro

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Provincial Mathematics Assessment at Grade 8Performance by Gender: Historical

f

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2011 Assessments

Timing

All elementary assessments will take place from May 26th to June 2nd, 2011. In response to teacher feedback, the Provincial Mathematics Assessment at Grades 3 and 5 will consist of three separate parts which can be administered on a flexible schedule at any time during the elementary administration schedule (please note: the content has not increased). It is expected that each section will be completed independently and students will not be permitted to return to a previous section. The general layout of the elementary assessments will be as follows:

Grade 3 Grade 5 Part A: Mental Math Multiple Choice

Part A: Mental Math Multiple Choice

Part B: Mental Math Multiple Choice

Part B: Mental Math Multiple Choice

Basic Facts (separate sheet) Basic Facts (separate sheet) Part C: Constructed Response

Part C: Constructed Response

The Provincial Mathematics Assessment at Grade 8 will take place on Monday, June 6th (calculator) and Tuesday, June 7th (non-calculator), 2011. The general layout of the assessments will be as follows:

Grade 8 Part A: Calculator Multiple Choice Constructed Response Part B: Non-Calculator Mental Mathematics Multiple Choice Constructed Response

Administration To assist teachers and to ensure consistent administration, all teachers will use the Teacher Guide in the administration of the May/June assessment. Every teacher will receive the Teacher Guide along with each class set of assessment booklets.

Answer Sheets (bubble sheets) are no longer required for the provincial mathematics assessments.



Marking

To align with the new curriculum and new assessment format, a rubric was developed to assist in scoring constructed-response questions. There are two forms of the rubric: a General Rubric, which can be applied to problem-solving responses in all grade levels (see below) and a Specific Rubric, which is an adaptation of the General Rubric and is tailored specifically to measure the competencies demonstrated in response to specific questions (see Appendix A). The new rubric replaces the ‘point system’ where information can get lost in conventions (i.e., calculation errors, using the incorrect or omitting units, etc.). The Rubric is diagnostic in that each competency is assessed individually within a range of proficiency.

Mathematics Rubric (general)

1 2 Knowledge Snapshot: Solution is partially complete Snapshot: Correct solution • Recalls factual and procedural

knowledge with few errors. • Generally demonstrates accurate

application of procedures. • May be minor errors or omissions but

do not detract from meaning.

• Recalls factual and procedural knowledge with accuracy.

• Consistently demonstrates accurate application of procedures.

• No errors or omissions.

Processes & Problem Solving

Snapshot: Some evidence of appropriate strategy

Snapshot: Efficient strategy implemented

• Evidence of drawing on some prior knowledge.

• Makes some connections between and within the different strands of mathematics.

• Successfully selects effective strategies to solve the given problem.

• Demonstrates evidence of thought process.

• May omit some aspects of problem. • Provides reasonable answer within

context of problem.

• Analyses situation and extends prior knowledge.

• Makes connections between and within the different strands of mathematics.

• Successfully selects efficient strategies to solve the given problem.

• Methodical and organized thought process.

• All aspects of problem addressed. • Recognize and extract the mathematics

embedded in the situation (mathematize).

• Provides reasonable answer within context of problem.

Communication Snapshot: Diagrams and explanation may contain errors or be incomplete.

Snapshot: Work is clear, detailed, organized. Diagrams and explanation clear.

• Generally uses mathematical language, units and symbolic notation.

• Uses some models and graphic representations to communicate thinking; some representations may lack clarity.

• Records and explains reasoning and procedures with some clarity.

• Demonstrates evidence of thought process.

• Some awareness of audience and purpose.

• Consistently uses precise mathematical language, units and symbolic notation.

• Uses models and graphic representations efficiently to represent and communicate with clarity.

• Records and explains reasoning and procedures with clarity, precision and thoroughness.

• Thought processes can be easily followed. • Awareness of audience and purpose.



Scoring Using the Rubric When marking a student response, teachers first assess the student’s knowledge and accuracy of a particular outcome; for example, a student’s ability to calculate area accurately. With a new lens, teachers then score the response based on the student’s ability to problem solve; to extract the necessary information from the scenario, select an appropriate strategy, and come to a reasonable solution. Finally, teachers then approach the same response regarding the student’s ability to communicate effectively; for example, using the appropriate units, models and mathematical language. Each competency is scored separately. The following section includes exemplars demonstrating this process.

1 2

Knowledge Snapshot: Solution is not fully complete Snapshot: Correct solution

• Decomposing 200 cm into 3 parts (not

necessarily equal) • Adding and subtracting 50 blue beads

and 30 red beads

• Recalls factual and procedural knowledge with few errors

• Generally demonstrates accurate application of procedures

• May be minor errors or omissions but do not detract from meaning

• Recalls factual and procedural knowledge with accuracy

• Consistently demonstrates accurate application of procedures

• No errors or omissions

Processes & Problem Solving Snapshot: Some evidence of appropriate strategy Snapshot: Appropriate strategy implemented

• Extracting necessary information from problem

• Designing/creating necklaces based on criteria

• Evidence of drawing on some prior knowledge • Makes some connections between and within

the different strands of mathematics • Successfully selects effective strategies • Demonstrates limited evidence of thought

process • May omit some aspects of problem • Provides reasonable answer within context of

problem

• Analyses situation and extends prior knowledge

• Makes connections between and within the different strands of mathematics

• Successfully selects efficient strategies to solve the given problem

• Methodical and organized thought process • All aspects of problem addressed • Recognize and extract the mathematics

embedded in the situation (mathematize) • Provides reasonable answer within context of

problem



• Diagrams

• Explanations

• Showing steps/calculations

• Generally uses mathematical language, units and symbolic notation

• Uses some models & graphic representations to communicate thinking; some representations may lack clarity

• Records and explains reasoning and procedures with some clarity. Demonstrates evidence of thought process

• Some awareness of audience and purpose

• Consistently uses precise mathematical language, units and symbolic notation

• Uses models & graphic representations efficiently to represent and communicate thinking

• Records and explains reasoning and procedures with clarity, precision and thoroughness so thought processes can be easily followed

• Awareness of audience and purpose

Exemplars The exemplars provided in Appendix A are a small sample of the items drawn from the pilot assessments. The problem-solving items are in the first year of development. As a consequence, the availability of samples is limited at this point. As the development continues, more samples and exemplars will be available for release.

The bullets in this section identify the elements to look for within the competency, specific to each question.

When determining if the student response should get a 1 or a 2, look at each section holistically. Don’t count bullets, determine generally which category the response best belongs (exemplars can help with this determination).



2011 Reporting As in previous years, constructed-response items will be marked by a committee of teachers selected from across the province. In advance of the marking session, committees of teachers, numeracy leads and learning specialists will review each of the constructed-response questions, rubrics and student responses to ensure the standards for assessment are consistent. The General Rubric will be used for the problem-solving questions and adapted to each question dependent upon the targeted competencies within each question. The same General Rubric is used for all grade levels. The General Rubric and established exemplars will be used in the marking sessions to ensure marking consistency. Examples of the marking process, using the General Rubric (adapting it to make it specific to each question) and exemplars are included in Appendix A. Common errors are noted anecdotally during the marking sessions and are reported along with the rubrics and exemplars in the annual Information Bulletin (Appendix B). Upon completion of the marking session, the booklets are sent for scanning and computer compilation. Early in the school year, districts and schools will be provided with provincial, district and school level data regarding strand and competency achievement scores. Additional Information Samples of the 2011 mathematics assessments are available on the Assessment and Evaluation portal site at: https://portal.nbed.nb.ca/tr/AaE/Pages/default.aspx

Further information regarding the Provincial Assessment Program can be found on the Assessment and Evaluation portal site located at: https://portal.nbed.nb.ca/tr/AaE/Pages/default.aspx The Administration of Provincial Assessments Protocols and Procedures (2010) provides detailed information on the administration of the Provincial Mathematics Assessments and can be found at: https://portal.nbed.nb.ca/tr/AaE/Other/Pages/default.aspx For information on Exemptions and Accommodations, please refer to Protocols for Accommodations and Exemptions (2010) which can be found on the portal at: https://portal.nbed.nb.ca/tr/AaE/Documents/Assessment%20Protocols%20for%20Accommodations%20and%20Exemptions.pdf

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Appendices





Grade 3 - Beaded Necklace Specific Rubric (2010)

1 2 Knowledge Snapshot: Solution is not fully complete Snapshot: Correct solution • Decomposing 200 cm into 3

parts (not necessarily equal) • Adding and subtracting 50 blue

beads and 30 red beads






• No errors or omissions.

Processes & Problem Solving Snapshot: Some evidence of appropriate strategy

Snapshot: Appropriate Strategy implemented

• Extracting necessary information from problem

• Designing/creating necklaces based on criteria

• Evidence of drawing on some prior knowledge

• Makes some connections between and within the different strands of mathematics

• Successfully selects effective strategies • Demonstrates limited evidence of

thought process

• Analyses situation and extends prior knowledge

• Makes connections between and within the different strands of mathematics

• Successfully selects efficient strategies to solve the given problem

• Methodical and organized thought process

• Recognize and extract the mathematics embedded in the situation (mathematize)

• Communication Snapshot: Diagrams and explanation may

contain errors or be incomplete. Snapshot: Work is clear, detailed, organized.

Diagrams and explanation clear. • Diagrams • Explanations • Showing steps/calculations








• Awareness of audience and purpose • No errors or omissions

Criteria

• Three necklaces • String divided into three reasonable

lengths (25 cm +); does not need to sum to 200 cm

• All 50 blue and all 30 red beads used

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Grade 3 Exemplars

Rationale

Score Knowledge • Accurate calculation of 200 cm of string (50 cm + 50 cm + 50 cm (can have left overs) • Accurate calculation of 50 blue beads (10 + 10 + 30) and 30 red beads (10 + 10 + 10) • No errors or omissions

2

Problem Solving • Extracted necessary information from problem • Selected effective strategy (implicit) • Methodical thought processes; all aspects of problem addressed • Reasonable solution

2

Communication • Explanation clear, precise and thorough • Correct units included

2



Rationale

Score Knowledge • Correct division (estimation implicit) • Did not deal with remainder but question stipulates that left-overs are permitted

(“minor errors or omission but does not detract from meaning”)

Low 1

Problem Solving • Connected measurement with number • Extracted from problem necessity to divide string into 3 pieces • Attempt at addressing problem (part of criteria met) • Limited evidence of thought process

low 1

Communication • Units omitted • Recording of reasoning, procedures too limited

0

Rubrics are helpful tools to assess responses demonstrating a range of proficiency. However, it is sometimes not clear where or how to draw a line between the categories; for example, judging a response as a 1 or a 2. Exemplars help to make these judgements. The following demonstrates how exemplars can be used for this purpose by highlighting that within each category there can be high and low 1s and 2s.



Grade 5 Birdhouse Specific Rubric (2010) 1 2 Knowledge Snapshot: Solution is not fully complete Snapshot: Correct solution - area calculation for large

sheet - area calculation for

birdhouse sides - multiplication/division for

number of birdhouse sides able to fit on sheet






• No errors or omissions



- attempt to find area of sheet - attempt to find area of

birdhouse sides - attempt to find number of

sides able to fit on sheet



• Successfully selects effective strategies • Demonstrates limited evidence of

thought process

• Analyses situation and extends prior knowledge • Makes connections between and within the different

strands of mathematics • Successfully selects efficient strategies to solve the

given problem • Methodical and organized thought process • Recognize and extract the mathematics embedded in

the situation (mathematize)

Communication

Snapshot: Diagrams and explanation may contain errors or be incomplete.


- use of diagrams - calculations/steps shown


• Uses some models and graphic representations to communicate thinking; some representations may lack clarity

• Records and explains reasoning and procedures with some clarity Demonstrates evidence of thought process



• Uses models and graphic representations efficiently to represent and communicate thinking



2 m (200 cm)

Area of Sheet: 2m x 1m = 200cm x 100cm = 20 000 cm2 Area of birdhouse 20cm x 20cm = 400 cm2 (one side) 400cm2 x 6 = 2400 cm2 (material required for one birdhouse) Number of birdhouses possible:

20 000cm2 ÷ 2400cm2 = 8 birdhouses (813

)

1 m (100 cm)

10 rows of 5 (20cm x 20cm) = 50 (20cm x 20cm squares)

50 ÷ 6 = 813



Grade 5 Exemplar

Rationale

Score Knowledge • Accurate area calculation for large sheet • Accurate calculation for area of each birdhouse side • Accurate calculation for required amount for all birdhouse sides

2

Problem Solving • Identified components of problem (area of sheet – are of birdhouse sides) • Employed efficient strategy • Arrived at reasonable solution

2

Communication • Use of diagrams and labels • Calculations shown • Correct units • Clarity – thought processes easily followed

2



Rationale

Score Knowledge • Accurate unit conversion of large sheet – cm to m • Did not know how to find area of large sheet (found perimeter) but found area of

one face of birdhouse sides (multiplied 20x20) • Found volume of birdhouse • Knew there are 6 faces in a cube

0

Problem Solving • Drew out problem (number of sides that could be cut from larger piece) • Some evidence of appropriate strategy for problem (attempted to find area of

sheet and attempted to find area of birdhouse faces) • Arrived at reasonable solution

1

Communication • Use of diagrams • Calculations and units shown • Lack of clarity in steps

1



Rubric Grade 8 – Shipping Container A.(2010)

1 2 Knowledge Snapshot: Solution is not fully complete

(surface area of at least one box) Snapshot: Correct solution

• Application of formula for surface area (implicit/explicit)

• Multiplication

• Recalls factual and procedural knowledge with few errors • Generally demonstrates accurate

application of procedures • May be minor errors or omissions

but do not detract from meaning

• Recalls factual and procedural knowledge with accuracy • Consistently demonstrates accurate application of

procedures • No errors or omissions



• Extract from the question the need to find surface area as opposed to volume



• Successfully selects effective strategies

• Demonstrates limited evidence of thought process

• Analyses situation and extends prior knowledge • Makes connections between and within the different

strands of mathematics • Successfully selects efficient strategies to solve the given

problem • Methodical and organized thought process • Recognize and extract the mathematics embedded in the

situation (mathematize)



• Diagram • Steps/calculations shown • Correct units









One Row of 12 cans (6 x 12 x 72)

Two rows of 6 cans (12 x 12 x 36)

Three rows of 4 cans (18 x 12 x 24)

12 cm

12cm

12 cm

36 cm

72 cm

24 cm

Surface Area: (12 x 6) x 2 = 144 cm2 (72 x 12) x 2 = 1728 cm2 (72 x 6) x 2 = 864 cm2 = 2736 cm2

Surface Area: (12 x 12) x 2 = 288 cm2 (36 x 12) x 4 = 1728 cm2 = 2016 cm2

Volume: (12 x 18) x 24 = 5184 cm3 Surface Area: (12 x 18) x 2 = 432 cm2 (24 x 12) x 2 = 576 cm2 (24 x 18) x 2 = 864 cm2 = 1872 cm2



Grade 8 Exemplars

Rationale

Score Knowledge • Accurate calculations for surface area for all three box shapes • No errors or omissions

2

Problem Solving • Correctly determined problem – need to find surface area for 3 different shapes • Appropriate strategy implemented • Methodical thought process • Reasonable solutions

2

Communication • Use of diagrams, labels • Correct units • Calculations shown • Clarity, awareness of audience

2



Rationale

Score Knowledge • Correct calculations for surface area • A: incorrectly calculated 4 sides at 72x12 (rather than 2 and 2 at 6x72) • B: correct calculations for surface area • C: incorrect value for length of side (6x4=18) carried calculation through

1

Problem Solving • Correctly determined problem – need to find surface area for 3 different shapes • Appropriate strategy implemented • Methodical thought process • Reasonable solutions

2

Communication • Use of diagrams, labels • Correct units • Calculations shown • Clarity, awareness of audience

2



Rubric Grade 8 – Shipping Container B. (2010)

Appropriate Strong Processes & Problem Solving Snapshot: Some evidence of appropriate

strategy Snapshot: Appropriate Strategy implemented

• Spatial understanding that volume of 12 cans is constant regardless of how they are arranged





• Analyses situation and extends prior knowledge • Makes connections between and within the

different strands of mathematics • Successfully selects efficient strategies to solve the

given problem • Methodical and organized thought process • Recognize and extract the mathematics embedded

in the situation (mathematize)



• Expressing ideas with clarity – may use diagrams/tables



• Records and explains reasoning and procedures with some clarity Demonstrates evidence of thought process






Volume: 1 can 339.292 cm3 12 x 339.292 = 4071.504 cm3

The following exemplars demonstrate how it is not always necessary to assess all three competencies:



Rationale Score

Problem Solving • Deconstructed problem • Spatial understanding that volume of 12 cans is constant regardless of how they

are arranged

2

Communication • Does not explain reasoning or procedures with thoroughness (reader must have

prior knowledge to understand response) • Lacks detail (volume calculation), clarity

1



Rubric Grade 8 – Shipping Container C. (2010)

Appropriate Strong Processes & Problem Solving Snapshot: one criteria identified Snapshot: More than one criteria identified • Compare and contrast

attributes of three boxes • Evidence of drawing on some prior

knowledge • Makes some connections between

and within the different strands of mathematics



• Analyses situation and extends prior knowledge • Makes connections between and within the

different strands of mathematics • Successfully selects efficient strategies to solve the

given problem • Methodical and organized thought process • Recognize and extract the mathematics embedded

in the situation (mathematize)



• Expressing ideas with clarity - may use diagrams/tables

• organisation









Criteria

- Shipping - Packing - Handling - Material costs (surface area) - Environmental costs (surface area)



Rationale Score

Problem Solving • Comparison/contrast between the different box shapes • Different criteria (shipping, packing, storing) used to compare

2

Communication • Work is clear, organised. Diagrams and explanation clear. • Logical organisation (use of table)

2



Provincial Mathematics Assessment at Grade 3 - 2010

Common Errors Common errors are a compilation of marking session teacher feedback and post-marking session reviews by exemplar selection committee members based on the statistical analysis of each question. This list has been kept as brief as possible to ensure effectiveness. General • Mathematical vocabulary emerged as a problem in both the multiple-choice questions and in the

constructed-response answers. • Students demonstrated a lack of shape and space knowledge; in particular, with regards to naming and

knowing the properties of shapes. Basic Facts • Generally well done. • Students were not looking at the sign and use the inverse operation. • Subtraction was not understood as well as addition. Mental Math • Many students used traditional strategies. • Students struggled with subtraction. • Many different strategies used – some not very efficient. Operations • Compensation while subtracting three digit numbers – students didn’t seem to know what to do with the

residual value. • 60 – 19 = 59 a frequently occurring error. • Students struggled with subtraction throughout the assessment. Problem Solving • When graphing, students did not label interval accurately (labelled between grid lines or inconsistent

intervals). • Explanations lacked precision and mathematical language. • Different components of the problems were ignored; students solved only one part of the problem. • Students need to be encouraged to explain their reasoning to extend beyond superficial explanations.

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Provincial Mathematics Assessment at Grade 5 - 2010 Common Errors

Common errors are a compilation of marking session teacher feedback and post-marking session reviews by exemplar selection committee members based on the statistical analysis of each question. This list has been kept as brief as possible to ensure effectiveness. General

• Students didn’t seem to know place value; specifically, they struggled identifying the difference between hundredths and thousandths.

• Students had great difficulty with questions targeting the visualization process standard; for example, any question requiring conceptualization the relative size and volume of a 3 dimensional object.

• Mathematical vocabulary (communication process standard) hindered performance; for example, product, parallelogram, regular, expression, value, etc.

Basic Facts • Often the inverse operation was performed; for example 8 ÷ 8 = 64. • Poor performance on the 0 and 1 multiplication facts.

Operations • Identifying the correct operation was seen as a problem, i.e., using the inverse operation. • Many students did not seem to know the meaning of ‘product’. • Knowledge of basic facts was an issue in all calculations. For example, in questions requiring division,

the mistakes were made in subtractions rather than the process of long division.

Decimals • Operations with decimals were not well done. Place value seemed to be the issue – decimals were

not lined up, zeros as place holders after the decimal were ignored and extra zeros were added for convenience, and numbers were manipulated for easy subtraction

Problem Solving • Students had difficulty finding area and distinguishing between area, perimeter and volume. • Students often did not show their thinking – explanations were often unclear or omitted. • Different components of the problems were ignored; students solved only one part of the problem. • Students need to be encouraged to explain their reasoning to extend beyond superficial explanations.



Provincial Mathematics Assessment at Grade 8 - 2010 Common Errors

Common errors are a compilation of marking session teacher feedback and post-marking session reviews by exemplar selection committee members based on the statistical analysis of each question. This list has been kept as brief as possible to ensure effectiveness. General

• The concept of surface area was not well understood. Students added dimensions and often confused surface area and volume.

• Probability was a weak area. Students tended to add fractions. Mental Math

• Compensation while subtracting three digit numbers – students didn’t seem to know what to do with the residual value.

• Students demonstrated a lack of place value knowledge and as a result frequently misplaced decimals.

Operations

• Knowledge of basic facts, in a broad sense, impeded all operations (integers, fractions, order of operations, percentages, rate, ratio, etc.).

• Order of operations – errors with adding and subtracting integers. Percentages

• Percentage responses were weak – students demonstrated a lack of conceptualization; for example, 5% was treated as 50% (5% of 28 = 14).

• 10% and 1% were not used as benchmarks; when doing so would have been a useful starting point. Fractions

• Many students demonstrated a lack of understanding of fractions in general. For example, conceptualizing/visualizing a fraction as a representation such as 45

÷ 2 was frequently recorded as 22.5

• Rather than leaving answers in the form of a fraction, quite often students converted fractions to

decimals incorrectly; for example 1018

was regularly converted to 1.8

• Operations with fractions: students frequently disregarded the distributive property; for example:

3 × 3 23

= 9 23

Problem Solving • Some students didn’t seem to know how to get started (lack of exposure?). • Students did not create diagrams to assist with problem solving solutions. • Many did not estimate to check the reasonableness of their answers. • Students need to be encouraged to explain their reasoning to extend beyond superficial explanations.