INSIGHTS FOR TEACHERS - NMSSA Mathematics and Statistics 2018 - National Monitoring Study of Student Achievement ...

 
INSIGHTS FOR TEACHERS - NMSSA Mathematics and Statistics 2018 - National Monitoring Study of Student Achievement ...
N MSSA Rep o r t 19- IN

INSIGHTS
FOR TEACHERS
NMSSA Mathematics
 and Statistics 2018
INSIGHTS FOR TEACHERS - NMSSA Mathematics and Statistics 2018 - National Monitoring Study of Student Achievement ...
NMSSA
                                                                                                       2018 Project Team                 EARU                         NZCER
                                                                                                       Management Team                   Sharon Young                 Charles Darr
                                                                                                                                         Albert Liau
             Wā n a n g a t i a t e P u t a n g a Ta u i r a                                                                             Lynette Jones
                                                                                                                                         Jane White
             National Monitoring Study
             of Student Achievement                                                                    Design/Statistics/                Alison Gilmore               Charles Darr
                                                                                                       Psychometrics                     Albert Liau                  Hilary Ferral
    Published on behalf of the Ministry of Education by Educational Assessment Research Unit (EARU),                                     Mustafa Asil                 Jess Mazengarb
    University of Otago, PO Box 56, Dunedin 9054, New Zealand.
                                                                                                       Curriculum/Assessment/            Jane White                   Linda Bonne
    https://nmssa.otago.ac.nz                                                                          Task development                  Sharon Young                 Jonathan Fisher
    NMSSA Report 19-IN: Insights for Teachers - NMSSA Mathematics and Statistics 2018                                                                                 Teresa Maguire
    ISSN: 2350-3238       ISBN: 978-1-927286-53-1                                                      Insights reporting                                             Linda Bonne
    Available online only at http://nmssa.otago.ac.nz                                                                                                                 Charles Darr
                                                                                                                                                                      Teresa Maguire
    © Crown 2020. All rights reserved.
    Images are copyright © Crown 2020 on pages 4, 10, 13, 28.                                          Programme Support                 Lynette Jones                Jess Mazengarb
                                                                                                                                         Linda Jenkins
    NMSSA is conducted by EARU and NZCER under contract to the Ministry of Education, New Zealand.                                       James Rae
                                                                                                                                         Fiona Rae
                                                                                                                                         Lee Baker
                                                                                                       External Advisors: Jeffrey Smith – University of Otago, Marama Pohatu – Te Rangatahi Ltd

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INSIGHTS FOR TEACHERS - NMSSA Mathematics and Statistics 2018 - National Monitoring Study of Student Achievement ...
CONTENTS
Part 1: The NMSSA mathematics and statistics assessment                 4
        What is NMSSA?                                                  4
        The 2018 NMSSA mathematics and statistics assessment            5
        The NMSSA mathematics and statistics scale                      6
        How did students do on the 2018 mathematics
        and statistics assessment?                                       6

Part 2: Insights about mathematics learning                             8
  Spatial reasoning: Introduction                                       10
  Insight 1: Using the language of spatial reasoning helps students
		           to communicate their thinking
		           Task: Moving Models                                        12
  Insight 2: Spatial reasoning involves visualisation and mental
		 manipulation
		           Task: Building in the Mind’s Eye                           16
  Fractions and percentages: Introduction                               22
  Insight 1: Students need to work with different models of fractions
		           Task: Fractions                                            23
  Insight 2: Understanding percentages begins early
		           Task: Clothes for Sale – Year 4
  Insight 3: Successful problem solving with percentages
                                                                        24
                                                                             The purpose of this report
		           requires students to identify and represent                     This report is designed to support the teaching of mathematics and statistics in primary
		           multiplicative relationships                                    and intermediate classrooms. It draws on insights generated from the assessment of
		           Task: Percentage Problems                                  26   the mathematics and statistics learning area carried out by the National Monitoring
                                                                             Study of Student Achievement (NMSSA) in 2018.
  Collaborative problem solving: Introduction                           29
             Task: Moving and Jumping                                   30   The report is in two main parts. Part 1 introduces NMSSA and the NMSSA assessment
                                                                             of mathematics and statistics. Part 2 presents practical insights relating to three focus
  Insight 1: Most students can collaborate on a task that is
                                                                             areas: spatial reasoning, fractions and percentages, and collaborative problem solving.
		           engaging and clearly structured                            34
  Insight 2: Working systematically and making links between
		           problems enhances task success                             35
  Insight 3: Thinking algebraically involves using a range
		           of problem-solving strategies                              37

Useful resources and references                                         41

                                                                                                                                                                         3
INSIGHTS FOR TEACHERS - NMSSA Mathematics and Statistics 2018 - National Monitoring Study of Student Achievement ...
PART 1
       The National Monitoring Study of Student Achievement
       mathematics and statistics assessment

    What is NMSSA?
    NMSSA is designed to assess student achievement across the New Zealand Curriculum (NZC) (Ministry of Education, 2007) at
    Year 4 and Year 8 in New Zealand English-medium state and state-integrated schools. Each year, nationally representative samples
                                                                                                                                       NMSSA
                                                                                                                                       Wā n a n g a t i a t e P u t a n g a Ta u i r a
    of students from 100 schools at each of these two year levels are assessed in one or more learning areas. The mathematics and      National Monitoring Study
    statistics learning area was assessed in 2013 and again in 2018.                                                                   of Student Achievement

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INSIGHTS FOR TEACHERS - NMSSA Mathematics and Statistics 2018 - National Monitoring Study of Student Achievement ...
The 2018 NMSSA mathematics
and statistics assessment
To assess the Mathematics and Statistics learning area in 2018, the NMSSA project
team developed a two-part assessment called the Mathematics and Statistics (MS)
assessment.
The first part was a group-administered paper-and-pencil assessment which
was completed by up to 25 students in each school. It was made up of a mixture
of multi-choice and short-answer questions. Students completed one of
14 assessment forms which were carefully linked together using common items.
The second part of the assessment involved several ‘in-depth’ tasks, which
were completed by up to six of the students who did the group-administered
assessment in each school. Students were videoed while they worked on these
tasks, usually in a one-to-one interview with a teacher assessor (TA). The tasks
included a focus on students explaining their mathematical thinking, describing
and applying problem-solving strategies, and collaborating with a partner.

                                                                                    5
INSIGHTS FOR TEACHERS - NMSSA Mathematics and Statistics 2018 - National Monitoring Study of Student Achievement ...
PART 1: The National Monitoring Study of Student Achievement mathematics and statistics assessment

    The NMSSA mathematics and statistics scale
    NMSSA used students’ responses to both parts of the 2018 assessment to construct a
    measurement scale–the mathematics and statistics (MS) scale1.
    The figure on the facing page shows the MS scale. The descriptors show how students’
    mathematics and statistics understandings increase in sophistication as the scale
    score increases and indicate the sorts of things students typically know and can do
    when they score at different parts of the scale. For example, a student scoring about
    110 units on the scale would typically be able to do the things described at that level.
    They would generally find the skills and knowledge described lower on the scale more
    straightforward. The descriptors further up the scale would typically involve knowledge                                                                        Year 4                 Year 8
                                                                                                                                                          180
    they were less clear about or skills they were unable to demonstrate consistently.
    It is important to note that the scale descriptors represent the knowledge and skills                                                                 160
    that were measured by the assessment items and are not intended to represent the
    mathematics2 learning area in its entirety.                                                                                                           140
                                                                                                                                                                                 Level

                                                                                                                                       scale score (MS)
                                                                                                                                                                                  4+
                                                                                                                                                          120
    How did students do on the 2018 mathematics                                                                                                                                 Level 3
    and statistics assessment?                                                                                                                            100

    The graph shows Year 4 and Year 8 students’ achievement on the 2018 NMSSA                                                                             80                    Level 2
    mathematics and statistics assessment, by levels of the curriculum.
    The 2018 study found that most Year 4 students (81 percent) achieved at or above                                                                      60                    Below-
    curriculum level 2, the expectation for the end of Year 4. In 2018, 45 percent of Year 8                                                                                    Level 2
    students achieved level 4 or above, the curriculum expectation for the end of Year 8.                                                                 40
    The study also found that there was no significant change in achievement for Year
                                                                                                                                                          20
    4 students between 2013 and 2018. For Year 8 students, however, there was a small
    statistically significant increase in average achievement.
                                                                                                                                                           0
                                                                                                                                                                Distribution of Year 4 and Year 8 students’ scores on
                                                                                                                                                                the 2018 Mathematics and Statistics (MS) scale

                                                                                                                                                            The blurred lines in the graph show the boundaries
                                                                                                                                                            between curriculum levels. The lines are blurred
                                                                                                                                                            to indicate the uncertainty involved in defining
                                                                                                                                                            precise boundaries.

    1
        For detailed information about the scale description, go to the report, https://nmssa.otago.ac.nz/reports/2018/2018_NMSSA_MATHEMATICS.pdf
    2
        In the remainder of this report, the term ‘mathematics and statistics’ has usually been shortened to ‘mathematics’ to support readability.
6       This in no way lessens the importance of statistics as part of the mathematics and statistics learning area.
INSIGHTS FOR TEACHERS - NMSSA Mathematics and Statistics 2018 - National Monitoring Study of Student Achievement ...
2018 NMSSA • Mathematics and Statistics scale descriptors

                              NUMBER AND ALGEBRA                            MEASUREMENT AND GEOMETRY                              STATISTICS

              180 •

                      Describe and apply efficient strategies to solve
                      word problems.
              170 •
                    Solve division problems with 2-digit divisors •
                    Find an unknown in an equation involving order
                    of operations • Solve a subtraction involving
                    mixed length decimals • Find a percentage of
              160 • a 3-digit number • Use proportional thinking to
                    solve percentage problems • Find fractions of a
                    fraction • Add and subtract fractions • Simplify
                    an improper fraction • Write an algebraic
                    expression to represent the nth term in a             Interpret a quadrilateral classification system
                    pattern.                                              • Explain how to convert between grams and
              150 •                                                       kilograms.

                    Plan, describe, and apply a strategy to solve
                    word problems • Solve division problems with
                    single•digit divisors • Find the square root of       Show a flexible understanding of perimeter, area
                    a whole number • Understand the order of              and volume • Identify direction and distance on a
              140 • operations • Add and multiply numbers involving       chart • Understand the effect of using a scale factor
                    one decimal place • Convert 3-digit decimals to       to enlarge a shape • Recognise invariant properties
                    percentages • Use proportional thinking to solve      under transformation • Rotate objects using turns
                    percentage problems • Find fractions of amounts       expressed in degrees.
                    • Order non-unit fractions • Recognise equivalent
                    fractions such as 12/16 • Add and subtract simple     Calculate the difference between two times several
              130 • fractions • Continue non•linear number patterns •     hours apart • Use measurements presented on a
                    Complete spatial number patterns.                     plan to find an unknown dimension • Use millilitres
                                                                          to estimate the capacity of a container • Read a
                    Recognise the number of 100s in a 4-digit

Scale score
                                                                          scale marked in 0.1 cm parts to measure length •
                    number • Order decimals up to 3 decimal places        Recognise properties of 2-D and 3-D shapes • Find
                    • Convert between simple fractions, decimals          the distance between two locations on a scale map
              120 • and percentages • Find a simple percentage of a       • Enlarge a shape by a scale factor.
                    number • Understand equality and inequality •
                    Continue number and spatial patterns.                 Use side by side scales to convert between inches
                                                                          and centimetres • Read scales to tenths • Work
                    Understand the effect of multiplying or dividing
                                                                          out the number of cubes in a shape where some
                    by 1 • Estimate simple multiplication problems •
                                                                          cubes are not visible • Identify which net can be
              110 • Use simple ratios to solve problems • Explain why
                                                                                                                                               PART 1: The National Monitoring Study of Student Achievement mathematics and statistics assessment

                                                                          used to make a given box • Use grid references and
                    a fraction is greater than another • Find a simple
                                                                          compass directions to show location • Recognise
                    fraction of an amount or set • Recognise different
                                                                          examples of different transformations.
                    representations of mixed fractions • Match a simple
                    number sentence with a word story • Continue
                    repeating spatial patterns with two variables.
              100 •                                                       Select appropriate units to measure a heavy object
                    Add and subtract 2-digit numbers • Solve simple       • Convert digital to analogue time • Read a half-way
                    multiplication and division problems • Estimate       mark on a scale marked in 20s • Have a sense of
                    the sum of two 3-digit numbers • Recognise which      the size of one metre • Draw a shape reflected in a
                    negative number is the lowest • Recognise the         mirror line • Identify movements on a grid • Identify
                                                                          cardinal compass directions.
               90 • number of tens in a 3-digit number • Recognise
                    simple fractions, decimals, and percentages •
                    Continue simple number and spatial patterns.          Recognise that a given container holds about
                                                                          one litre • Identify common shapes • Count the
                                                                          number of cubes in a shape where part of each
                                                                          cube is visible • Recognise a side view from a 3-D
               80 •                                                       representation.

                    Add groups of 10 • Understand the effect
                    of adding 0 • Write numerals as words •
                    Demonstrate a sense of place value up to 3
               70 • digits • Represent a simple fraction as an area •
                    Continue a repeating pattern with one variable.

                              NUMBER AND ALGEBRA                            MEASUREMENT AND GEOMETRY                              STATISTICS
         7
INSIGHTS FOR TEACHERS - NMSSA Mathematics and Statistics 2018 - National Monitoring Study of Student Achievement ...
PART 2
           Insights about mathematics learning

                                                                                                Spatial reasoning
                                                                                                Insight 1: Using the language of spatial reasoning helps students to communicate
    This section presents a series of insights about mathematics learning, based on students’
                                                                                                           their thinking
    responses to tasks included in the in-depth component of the NMSSA assessment.
    The insights relate to three focus areas.                                                   Insight 2: Spatial reasoning involves visualisation and mental manipulation

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INSIGHTS FOR TEACHERS - NMSSA Mathematics and Statistics 2018 - National Monitoring Study of Student Achievement ...
Fractions and percentages                                                  Collaborative problem solving
Insight 1: Students need to work with different models of fractions        Insight 1: Most students can collaborate on a task that is engaging
Insight 2: Understanding percentages begins early                                     and clearly structured

Insight 3: Successful problem solving with percentages requires students   Insight 2: Working systematically and making links between problems enhances
           to identify and represent multiplicative relationships                     task success
                                                                           Insight 3: Thinking algebraically involves using a range of problem-solving strategies

                                                                                                                                                                    9
INSIGHTS FOR TEACHERS - NMSSA Mathematics and Statistics 2018 - National Monitoring Study of Student Achievement ...
PART 2: Insights about mathematics learning: Spatial reasoning

     SPATIAL REASONING

      Introduction

     Spatial reasoning involves being able to visualise a 2- or 3-dimensional object and
     then mentally manipulate it. If you’ve ever tried to parallel park your car, played a
     game of Tetris or used a paper map to navigate, you’ve had to use spatial reasoning
     skills.
     Spatial reasoning includes both mental rotation and spatial orientation.Mental
     rotation is when we think about objects in our mind and move them around, such
     as when we imagine how a room might look if we shifted the furniture. Spatial
     orientation is when we think about the position of an object in relation to our own
     position, such as when we follow maps or complete mazes.

     Why is spatial reasoning important?
     Over the last few decades, there has been an increasing number of studies
     demonstrating that early ability in spatial reasoning is a powerful predictor of
     students who enjoy, enter and succeed in Science, Technology, Engineering, and
     Mathematics (STEM) disciplines. People working in STEM areas often use diagrams,
     maps and models to represent ideas when working through tasks or problems.

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PART 2: Insights about mathematics learning: Spatial reasoning

Where does spatial reasoning fit in the
curriculum?
Spatial reasoning is most visible in the Geometry and Measurement
strand of the Mathematics and Statistics learning area under the
sub-strands ‘Position and Orientation’ and ‘Shape’. Geometry is described
in the curriculum:
          Geometry involves recognising and using the properties and
          symmetries of shapes and describing position and movement.
                                                 (Ministry of Education, 2007, p.26)

The NZMaths website3 explains that we teach geometry because ‘we live in
an obviously three-dimensional world that we walk through, explore and
use every minute of every day’. We need to understand this world ‘in order
to carry out even the simplest of tasks’.

3
    https://nzmaths.co.nz/geometry-information
                                                                                       11
PART 2: Insights about mathematics learning: Spatial reasoning

      Insight 1:
      Using the language of spatial reasoning helps students to communicate their thinking

      Task: Moving Models
      Moving Models was an in-depth task given to both Year 4 and Year 8 students to assess
      their ability to mentally manipulate objects.
                                                                                                    How did the students respond?
                                                                                                    Student responses were categorised according to how many sets of instructions they
      In Part 1 of this task, a 3-dimensional model was placed in front of the student.
                                                                                                    needed to give the TA to succeed with the task.
      An identical model, in a different orientation, was placed in front of the teacher assessor
      (TA).                                                                                         Results on the next page show how the students were scored on Parts 1 and 2 of
                                                                                                    the Moving Models task, by year level. Students who scored a ‘2’, the highest scoring
      The student was asked to tell the TA how to move the student’s model, so it looked like
                                                                                                    category, were more likely to use specific spatial language that enabled them to give
      the TA’s model. The students were not permitted to touch the models; however, they
                                                                                                    clear instructions.
      could point or make gestures as they gave their instructions. After repeating their
      instructions back to them, the TA then followed the student’s instructions, checking
      at the end to see if the models looked the same. If the models did not look the same,
      students were permitted to give additional instructions. Part 2 of this task involved a
      larger and more complex model.

     PART 1                                                                                              PART 2

                         Student ’s model                            TA ’s model                                             Student’s model                         TA’s model

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PART 2: Insights about mathematics learning: Spatial reasoning

                                                                                                                                          % response

                                                                 Results for Moving Models, by year group:                                Y4 Y8
                                                                   PART 1

                                                                   Scoring category                                          Descriptor
                                                                   2                     Gives one set of instructions that allows the TA to
                                                                 		              successfully move student’s model to look like TA’s model 64 73
                                                                   1                         Gives a set of instructions that does not initially
                                                                 		                    allow the TA to successfully move student’s model to
                                                                 		             look like TA’s model; after TA prompt, then gives additional
                                                                 		                        instructions which do allow the TA to successfully
                                                                 		                             move student’s model to look like TA’s model 30 23
                                                                         0                                           Unable to complete    6     4

                                                                   PART 2
                                                                   2                     Gives one set of instructions that allows the TA to
                                                                 		              successfully move student’s model to look like TA’s model 59 81
                                                                   1                         Gives a set of instructions that does not initially
                                                                 		                    allow the TA to successfully move student’s model to
                                                                 		             look like TA’s model; after TA prompt, then gives additional
                                                                 		                        instructions which do allow the TA to successfully
                                                                 		                             move student’s model to look like TA’s model 30 17
                                                                         0                                           Unable to complete 11       2

                                                                                                                                                       13
PART 2: Insights about mathematics learning: Spatial reasoning

      Students who scored 2
     In the four examples below, students have given one set of instructions that allowed the TA
     to successfully move the student’s model to look like the TA’s model.

                                                                                                                     Spatial language
                          Flip it up vertically and then                       Rotate it so it’s standing up         Students who scored a 2 often used specific
                         give it a rotation of 90 degrees                   and then rotate like a quarter turn      spatial language words such as clockwise,
                                   anticlockwise.                                     anticlockwise.                 anticlockwise, vertical, horizontal, degrees,
                                                                                                                     rotate, quarter turn.

                                               Student’s model                                          TA’s model

                                                                                    Can you flip it up so it’s       Spatial language
                          Put it on its triangle bottom.                     standing on the three blocks and        Some Year 4 students used less specific
                            Turn it to the pointy end                       the one block on the side then turn      spatial language words such as flip and
                          – this pointy end facing me                        it so it’s facing me – the one block    pointy.
                                                                                           is facing me.

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PART 2: Insights about mathematics learning: Spatial reasoning

     Students who scored 1
 In this example the student gives a set of instructions that does not initially lead
 to making the student’s model look like the TA’s model.
                                                                                                 Literacy in mathematics -
 The student uses a lot of gestures to indicate how the TA should move the model.
                                                                                                 the language of spatial reasoning
 The TA has to seek clarification as the student does not use specific spatial language.         Spatial language is the explicit language used to describe position,
                                                                                                 shape, direction and transformation. There are five categories of
                                                                           Student               spatial language:
         Give me instructions to tell me how to move your                                           • Shape terms - standard names for objects: circle, square,
         model so it looks like my model.                                                             cone, pyramid, octagon, sphere, cylinder, cuboid,
                                                                                   Face it up.        tetrahedron, equilateral, scalene triangle
TA
         What do you mean face it up?                                                               • Dimensional adjectives - words describing the size of
                                                       Make that bit [points to the cube that
                                                                                                      objects and spaces: big, little, long, short, tall, tiny, huge
                                                   is currently touching the table] point up.
                                                                                                    • Spatial features - words describing the features of objects
         So make that bit point up?                                                                   and spaces: edge, corner, line, curved, straight, flat,
            OK. Anything else?                                                                        angle, acute, congruent, parallel, equivalent, perimeter,
                                                                                          No.         circumference,

                                                                                                    • Spatial location and directions - words describing the
                          So you want me to face that bit up.                                         relative position of objects, people and points in space:
             [TA follows the student’s instructions and moves the model.]                             between, in, on, under, above, below, forward, behind, near,
                            Does your model look like mine?                                           far, middle, top

                                                                                          No.       • Transformational language - words describing the
                                                                                                      movements of objects and people: turn, slide, flip, translate,
         Tell me what else I need to do to make                                                       rotate, clockwise, anticlockwise, degrees.
         your model look like mine.
                                                                              Turn it around.    Exposing children to spatial language from a young age is very
         Turn it around. OK. How will I turn it around?                                          important as it has a strong influence on the development of spatial
                                                                                                 cognition and mathematical skills.
                                                  [Student uses hands to indicate the model      Teachers can help students to develop their spatial language by
                                                  should be turned 90 degrees towards her.]      using mathematical vocabulary when describing objects, locations
                                                                                                 or transformations.
         The TA turns the model and the two models are now in the same orientation.

                                                                                                                                                                        15
PART 2: Insights about mathematics learning: Spatial reasoning

     Insight 2:
     Spatial reasoning involves visualisation and mental manipulation

     TASK: Building in the Mind’s Eye                                                          How did students respond?                                                   % correct
                                                                                                                                                                           responses

     Building in the Mind’s Eye was another in-depth task given to both Year 4 and Year 8      Overall, Year 8 students performed better than Year 4 students on these    Y4 Y8
     students to assess their ability to mentally manipulate objects.                          tasks, with both groups finding the second and third tasks considerably
                                                                                               more challenging than the first task.
     In this three-part task, students listened to step-by-step descriptions of how to
     construct 3-dimensional models using plastic cubes. After hearing each description,           Results for Building in the Mind’s Eye, by year level
     they were then shown four photographs and asked to select which one looked like
                                                                                               		                                                                  Part 1 65 79
     the picture they had built in their mind’s eye. Students were generally successful when
     the model they were asked to construct was straightforward and involved only one          		                                                                  Part 2 27 43
     transformation. More complex instructions involving several construction steps or         		                                                                  Part 3 35 47
     transformations resulted in more challenge for students.

     PART 1

              Imagine you have four cubes.
              Three are black and one is white.
                                                                                                                                                     The 3-dimensional shape
              Take the three black cubes. Snap them
                                                                                                                                                     described here is made of
              together in a row and lie them down flat
              in front of you.
                                                                                                                                                     only four cubes.
              Picture what this looks like in your mind.                                                                                             One simple transformation
              Now put one white cube on top of the middle                                                                                            is performed on the
              black cube.                                                                                                                            shape once it has been
              Picture what this looks like in your mind.                                                                                             constructed.
              Now flip the shape upside down.
              Put a picture of this in your mind.

                                                                                                                         Correct response

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PART 2: Insights about mathematics learning: Spatial reasoning

PART 2

         Imagine you have six cubes.
         Four are white and two are black.
         Take the four white cubes. Snap them together
         in a row and lie them down flat in front of you.
         Picture this in your mind.
                                                                                    The 3-dimensional shape
         Now snap one black cube on top of the                                      described here is made of
         far left white cube of the row.
                                                                                    six cubes.
         Picture what this looks like in your mind.
                                                                                    Two transformations are
         Now put the other black cube next to the
         first black cube, on top of the next white cube                            performed on the shape
         in the row.                                                                – one half turn and one
         Picture what this looks like in your mind.
                                                                                    quarter turn.
         Now turn the shape to the left a half turn
         or 180 degrees.
         Picture this in your mind.
         Now flip the shape up a quarter turn                    Correct response
         to the right.
         Put a picture of this in your mind.

                                                                                                                17
PART 2: Insights about mathematics learning: Spatial reasoning

     PART 3

              Imagine you have eight cubes. Four are black,
              two are white and two are blue.
              Snap a blue cube and a black cube together. Put
              them down in front of you with the black cube
              on the right.
              Picture this in your mind.
              This time you are going to build a tower shape.
              Now snap the other blue cube to another black
              cube, this time with the blue cube
              on the right.                                                              The 3-dimensional shape
              Put this second pair of cubes on top of the first                          described here is made of
              pair with the black cubes on top of each other                             eight cubes.
              and the top blue cube facing to the right.
                                                                                         The cubes are placed
              Picture this in your mind.
                                                                                         in a range of different
              Now put one white cube on top of the                                       orientations – left, right,
              black cubes.
                                                                                         behind, on top of.
              Picture this in your mind.
                                                                                         No transformations are
              Snap the remaining white cube on to the back
                                                                                         performed on this shape.
              of the first white cube so it is facing away            Correct response
              from you.
              Now put a picture of this in your mind.
              Take one of the remaining black cubes
              and put it on top of the white cube that
              is closest to you.
              Put a picture of this in your mind.
              Lastly, take the final black cube and snap
              it on to the front of the black cube you
              just placed so that it is facing towards you.
              Now picture this in your mind.

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PART 2: Insights about mathematics learning: Spatial reasoning

What can teachers do?
How can I include spatial reasoning in my classroom programme?
For some years there was an assumption that spatial reasoning skills were innate –
                                                                                          Introduce challenges that develop students’ visualisation skills
you could either parallel park or you couldn’t. Recent research, however, tells us that
spatial skills are malleable and can be taught.                                            • The following quick challenges can help students develop spatial visualisation
Here are some things to consider:                                                            skills:
Start early                                                                                   »» Can you draw this? Students are given a piece of paper with the outline of
                                                                                                 a square on it. They are then shown a geometric design drawn within the
   • Helping students develop their spatial understanding can start from a very
                                                                                                 boundaries of a similar square (see examples below). After 10 seconds of
      young age. Te Whāriki (Ministry of Education, 2017) suggests that to develop
                                                                                                 viewing time, the students attempt to re-create the exact design in their own
      spatial understanding in the early years, students should have opportunities
                                                                                                 square.
      to fit things together, take things apart, rearrange and reshape objects
      and materials, look at things from different viewpoints and use magnifying
      glasses. It suggests young children should also have the chance to construct
      3-dimensional structures such as making models from pictures. When students
      start primary school, they will bring with them a range of experiences which
      teachers can build on by extending their geometric concepts and vocabulary.
   • Reading books and watching videos that use spatial language such as positional
      words, are other ways for teachers to explore and extend these concepts and
      vocabulary (also see p.15, Literacy in mathematics – the language of spatial            »» Can you build this? The same process is used as for Can you Draw this? except
      reasoning).                                                                                students are shown a geometric structure made from multi-link blocks for
                                                                                                 10 seconds and then asked to re-create it using their own set of multi-link
Encourage spatial play
                                                                                                 blocks.
   • Building with blocks, doing jigsaw puzzles, tangrams or using pentominoes
     give students opportunities to develop spatial skills. Researchers found that            »» Shape transformer: Similar to a ‘Function machine’ activity where two columns
     children who had spent a lot of time playing with blocks, puzzles and board                 of numbers are shown – input and output - and a rule is developed to explain
     games scored higher on tests of spatial ability.                                            the function used to change the input into the output, for example, add 3 or
                                                                                                 multiply by 2. In this version a shape is drawn into the Input column and a
   • Many board games are available now that require students to use spatial                     transformed shape is drawn in the output column. Students have to work out
     thinking. For example, Blokus, Patchwork, Cottage Garden, Indian Summer,                    what transformation has occurred, e.g. rotation 45o or translation, and predict
     Nmbr9, Tetris and Jenga. Many board games have the added advantage of                       what will happen to future shapes.
     needing addition and multiplication skills to calculate scores. There are also
     many social skills, such as turn taking, that can be practised while playing board       »» Barrier games: Students sit on either side of a barrier (cardboard folder, book
     games.                                                                                      or similar). One student builds a shape or figure using blocks or multi-link
                                                                                                 cubes then describes it to their partner who then builds the same figure.
                                                                                                 Students compare structures and swap roles.

                                                                                                                                                                                   19
PART 2: Insights about mathematics learning: Spatial reasoning

          »» Building in the Mind’s Eye, as described in this Insight. Students are given oral             • Several activities that can also help with spatial visualisation are mental folding,
             instructions to build a 2-dimensional or 3-dimensional figure in their mind. They               paper folding and working with nets. Mental folding involves students being
             are then shown a series of pictures and asked to identify the one that matches                  shown a picture of a shape and asked to imagine folding it along a dotted line.
             what is in their mind. The instructions can be made more complex by adding                      They select which of four images represents what the shape would look like after
             more cubes; adding more instructions; adding more positional language (on                       being folded. Similarly, paper folding requires students to imagine the result
             top of, behind, left, right) or adding more transformational language (rotate,                  of folding and cutting a piece of paper then unfolding it. Activities that require
             flip, slide, turn). Students could work in pairs or groups and develop their own                students to work out which nets fold to make shapes are further examples of
             versions of this task.                                                                          tasks that involve visualisation.

                MENTAL FOLDING                                            PAPER FOLDING                                                                    MATCHING NETS II
          (from Hodgkiss, Gilligan, Tolmie,                   (from Ontario Ministry of Education, 2014)                        (an ARB activity: https://arbs.nzcer.org.nz/resources/matching-nets-ii)
              Thomas & Farran, 2018)

20
PART 2: Insights about mathematics learning: Spatial reasoning

Help students develop perspective-taking                                                    Use maps

   • Activities that encourage perspective-taking can help students to develop                • Using physical maps is another good way to develop spatial reasoning skills in
     spatial rotation skills. Set up blocks and shapes in a ‘scene’. Add a ‘photographer’       students. When using maps, there are opportunities to develop spatial language
     – a figurine holding a camera, ‘looking’ at the scene. Ask students to select              (left, right, between, etc.) and perspective-taking. Drawing maps, giving directions
     which of four photographs show what the ‘photographer’ would have seen.                    using a map and locating ‘hidden treasure’ are activities that can help develop
     Alternatively set up a scene with two different ‘photographer’ figurines                   these skills. [Refer to Social Studies Insights, p.73 for information on how students
     looking at the same shapes from different viewpoints. Show a photograph of                 performed on a map-based perspective-taking task.]
     one perspective and ask students which ‘photographer’ took the photo.

         (from Hodgkiss et al., 2018)

   • The Moving Models task, as described in this Insight, can easily be duplicated in
     the classroom and can help develop perspective-taking. Students could work
     in pairs or groups and develop their own versions of this task.

                                                                                                                                                                                        21
PART 2: Insights about mathematics learning: Fractions and percentages

     FRACTIONS AND PERCENTAGES

      Introduction

     Year 4 to Year 8 is a critical time for students to develop a strong grasp of fractions,
     decimals and percentages so they can apply them across the curriculum (not just in
     mathematics and statistics), move on to new mathematics learning, and make use of
     them in life. Research suggests that developing a strong conceptual grasp of these
     ideas is challenging and that it takes time. The focus of the tasks we describe in this
     section is fractions and percentages.

     Fraction task examples:

           1. Colour in        of this shape.                                                   4. Mark where    1   would be on this number line.

                                                                                                        0                                            2

           2. Colour in a        of these circles.
                                                                                                5. Colour in    of these circles.

           3. Draw a picture to show         1                                                  6. What is     as a decimal?        0.8

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PART 2: Insights about mathematics learning: Fractions and percentages

Insight 1:
Students need to work with different models of fractions

Fractions come in different guises and can carry a range of meanings. Students need                                                                          2018 (13) 2018 (13)
multiple opportunities to work with different models of fractions.                                                                                            % correct responses
                                                                                                                                                             Year 4      Year 8
TASK: Fractions                                                                           Results for Fractions task, 2018 (and 2013) by year level
One NMSSA task called Fractions asked students at both year levels to complete
                                                                                      		                                           Fractions questions
six questions focused on using different representations of fractions, shown on the
previous page. (The exact items are not shown here because we intend to use them      		                                  Colour in a fraction of an area    76 (69)     97 (95)
in future assessments. Instead, we show similar items.)
                                                                                      		                                Colour in a unit fraction of a set   36 (29)     80 (78)
How did the students do?                                                              		                      Draw a diagram to show a mixed fraction        15 (13)     71 (67)
The percentages of Year 4 and Year 8 students who responded correctly to each
question are shown in the table. Versions of the same questions were also asked in    		                       Show a mixed fraction on a number line        14 (13)     68 (68)
2013, allowing a comparison with students’ responses in 2018.
                                                                                      		                           Colour in a non-unit fraction of a set    23 (21)     70 (67)

                                                                                      		                         Write a non-unit fraction as a decimal        2 (1)     46 (41)

                                                                                      The improvement between Year 4 and 8 indicates the amount
                                                                                      of learning that goes on between the year levels. However, the
                                                                                      evidence suggests that, overall, students have a stronger grasp
                                                                                      of some ways of using fractions than others.

                                                                                                                                                                                    23
PART 2: Insights about mathematics learning: Fractions and percentages

     Insight 2:
     Understanding percentages begins early
                                                                                                       Clothes on Sale Card 1

     Task: Clothes on sale – Year 4
     The Clothes on Sale task asked Year 4 students to show their understanding of two           PART 1: 50% off the price of this T-shirt.
     commonly occurring percentages: 50% and 25%.
     The task was based around a shopping scenario and was presented by a teacher
     assessor (TA) as part of a one-to-one interview. The first part of this task focused on a

                                                                                                              $20
     discount of 50 percent and the second, a discount of 25 percent. For both parts of the
     task, students were asked what they thought the percentage meant, what the answer
     might be, and how they worked out the answer.

                                                                                                      Voucher 1                                  Voucher 2

                                                                                                       Clothes on Sale Card 2
                                                                                                                              VOUCHER
                                                                                                                                  ONE USE ONLY

                                                                                                                TAKE 50% OFF THE                      TA
                                                                                                                 REGULAR PRICE                         R
                                                                                                              $16
24
                                                                                                      Voucher 1                                  Voucher 2
PART 2: Insights about mathematics learning: Fractions and percentages

                                                                                                    $20

                                                                                              Clothes on Sale Card 2

How did the Year 4 students do?
Slightly over half of the Year 4 students were able to correctly discount the T-shirt    PART 2: 25% off the price of a hoodie.
by 50 percent. Almost all of these students were able to explain how they found the
answer.
Twenty-two percent were able to correctly discount the hoodie by 25 percent. In

                                                                                                    $16
response to the question, “What does 25% mean?”, 31 percent of the Year 4 students
explained that it meant one quarter.
Although achievement objectives in the NZC don’t explicitly mention percentages
before level 3, most students in Year 4 will be exposed to percentages in their daily
lives. They will also come across them in a range of curriculum contexts.
The success of many Year 4 students on this task suggests that there are opportunities
                 Voucher
for teachers to draw on these 1real-life understandings when teaching ideas about             Voucher 2
fractions and percentages. It is also important to remember that students will not
automatically understand percentage because of their real-life experiences alone.

                                         VOUCHER                                                                     VOUCHER
                                             ONE USE ONLY                                                                ONE USE ONLY

                           TAKE 50% OFF THE                                                             TAKE 25% OFF THE
                            REGULAR PRICE                                                                REGULAR PRICE

                                                                                                                                        25
                 Voucher 1                                                                    Voucher 2
PART 2: Insights about mathematics learning: Fractions and percentages

                Insight 3:
                Successful problem solving with percentages requires students to identify
                and represent multiplicative relationships

                TASK: Percentage Problems
                Year 8 students were also administered a series of problems involving percentage
                questions as part of a one-to-one interview with a TA. The questions demanded strong
                conceptual understanding and went well beyond more routine questions that involve
                a percentage increase or decrease. Two examples of the problems are below.

Card 2                              Percentages Card 1                                                 Percentages Card 2                                Percentages Card 1

                                    A shop has a ‘Buy one get the second one                                                                             A shop has a ‘Buy one get the sec
 in a 25% off sale.                 at half price’ sale.                                               This T-shirt is in a 25% off sale.                at half price’ sale.
e is $30.                           If you choose 2 items that are the same price,                     The sale price is $30.                            If you choose 2 items that are the
st before it was put in the sale?   what percentage of the total cost will you save                    What did it cost before it was put in the sale?   what percentage of the total cost w
                                    altogether?                                                                                                          altogether?

       26
PART 2: Insights about mathematics learning: Fractions and percentages

How did the Year 8 students do?
The proportion of students who were able to solve each problem is shown below.             Drawing a picture or a diagram is a useful problem-solving approach with fractions
Students who answered correctly, generally demonstrated a strong understanding of          and percentages. Here is a student’s drawing and an explanation of how they solved
what the questions were asking and could explain the problem diagrammatically or           the “Buy one, get the second one at half price” problem.
as a series of multiplicative operations. There was evidence that some students had
an appreciation of the percentages involved but struggled to comprehend what the
question was asking and/or how to represent the proportional relationships involved.
Answering percentage problems such as these, requires students to have a strong
understanding of how the quantities in the questions relate to each other. The key
relationships have a multiplicative dimension and defining these is usually the key
to unlocking the problem. Developing the ability to think through these kinds of
problems goes well beyond applying a memorised procedure. Students need plenty
of opportunities to work on these kinds of problems, represent them in different ways,
and communicate how they might be solved.                                                                                                If you buy one you get another
                                                                                                                                          one but it’s only half the price.
                                                                                                                                           That means you pay for 1.5
                                                                               % correct                                                     instead of paying for 2.
                                                                               responses
Results for Percentage Problems, Year 8                                                                                                  The 0.5 you save is like a quarter
                                                                                  Y8                                                       of 2 – there are 4 halves in 2.
Problem                                                                                                                                     That means you save 25%.
number		        Problem text
  1           Calculate the percentage saved when buying two identical
		               items at a ‘Buy one, get the second one at half price’ sale      43
  2                              Calculate what a T-shirt with a sale price
		                   of $30 cost before it was put in a 25-percent-off sale       23

                                                                                                                                                                                27
PART 2: Insights about mathematics learning: Fractions and percentages

     What can teachers do?
     Fractions                                                                               Percentages
       • Provide students with multiple opportunities to work with different                   • Introduce simple percentages early, e.g., 50% and 25% can be included in
         representations of fractions, including a fraction as part of a set and an area,        meaningful contexts when fractions are also used.
         and as a number on a number line. Help them to make strong connections
         between these different representations. ‘Think boards’ are a resource that can       • Use pictures, models and materials to help students build understanding
         help build connections among different models. Think boards typically have              of what percentages mean and how they can be shown.
         four quadrants to show four models of the same problem or mathematical
         idea, like this one:                                                                  • Encourage students to visualise what a percentage means. Encourage them
                                                                                                 to use drawings or diagrams to represent the relationships in a percentage
                                      Story                                                      question. Make explicit connections between their understanding of fractions
                                                                                                 and decimals to consolidate meaning.
          Number Sentences

                                                                      Concrete Materials

                                      Picture

         A think board could equally be used to make connections between four ways
         of showing ¾ as: part of a rectangle, part of an array of 12 stars, a location on
         a number line (marked at one end with ‘0’ and at the other end with ‘1’), and in
         a story that involves ¾. In this case ¾ would be written in the rectangle at the
         centre of the board.
       • Provide learning experiences that involve unit fractions, non-unit fractions,
         and mixed fractions, shown in the ways listed above.
       • Emphasise fractions as parts of wholes, and that the size of ½ can vary
         according to the whole (½ of a small cake compared with ½ of a large cake).

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PART 2: Insights about mathemat

COLLABORATIVE PROBLEM SOLVING

 Introduction

What is collaborative problem solving?                                                How does collaborative problem-solving link to the NZC?
Collaborative problem solving involves working with other people to solve problems.   In the NZC, the disciplines of mathematics and statistics are introduced as: “related but
It has two aspects: a cognitive aspect and a social aspect.                           different ways of thinking and of solving problems” (p. 26). In the mathematics and
                                                                                      statistics learning area, achievement objectives at all levels are prefaced with:
1. The cognitive aspect involves:
                                                                                             In a range of meaningful contexts, students will be engaged in thinking
   • task regulation–problem analysis; goal setting; collecting information
                                                                                             mathematically and statistically. They will solve problems and model
     and adapting a strategy as more information becomes available; working
                                                                                             situations that require them to: …
     systematically and monitoring progress
                                                                                      Working with others to collaboratively solve problems is a very natural fit for
   • knowledge building–recognising patterns and relationships within the data,       mathematics. Collaborative problem solving, however, extends across the
     including cause and effect; hypothesising and testing hypotheses.                curriculum and can be directly related to the Key Competencies of Thinking, and
2. The social aspect involves:                                                        Relating to Others. Thinking includes problem solving by ‘using creative, critical,
                                                                                      and metacognitive processes to make sense of information, experiences, and ideas’
   • participating–being active; responding to and coordinating with others;
                                                                                      (Ministry of Education, 2007, p.12). Relating to Others ‘includes the ability to listen
     persevering to complete a task; sharing a sense of responsibility
                                                                                      actively, recognise different points of view, negotiate, and share ideas. … By working
   • perspective taking–integrating others’ contributions into one’s own thoughts     effectively together, [students] can come up with new approaches, ideas, and ways of
     and actions; showing responsiveness to the needs of a partner and the task       thinking’ (ibid).
   • social regulation–being aware of one’s own strengths and weaknesses, and
     those of one’s collaborators; being able to negotiate and reconcile different
     perspectives.

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PART 2: Insights about mathematics learning: Collaborative problem solving

     TASK: Moving and Jumping
     Students’ collaborative problem-solving skills were the focus of a task called Moving      To promote a collaborative approach, students were told that only one person could
     and Jumping. The task was presented to a pair of students by a teacher assessor (TA).      move the red counters and only one person could move the blue counters. The TA
     Each pair was randomly selected from the group of students involved in the NMSSA           instructed the students: ‘You cannot move each other’s counters. Show me how you
     programme at each school (one pair per school). Whether or not the students knew           can work together, and talk to each other, to solve this problem.’
     each other or had experience working together was not taken into consideration
                                                                                                When students successfully solved a problem, they recorded the number of moves
     when selecting the students. The students’ work on the task was videoed for later
                                                                                                required in a table that was provided.
     analysis.
     The task involved a series of five problems focused on a simple, counter-based activity.
     The activity involved swapping the positions of two groups of counters positioned at
     either end of a simple grid. Students were given the following instructions:

      There are two ways to move a counter, by moving ahead one
      square or by jumping a counter of the other colour.
      A counter cannot move backwards along the line.
      You cannot have two counters in the same square.
                                                                                                           B                   B                           R              R
      If you and your partner get stuck, just start again.
                                                                                                   The board set up for problem 1: the 2-counter problem

     In the first problem (see figure right) students needed to work out how many moves
     would be required to swap the position of the two blue and two red counters. Problems
     2 and 3 involved sets of three and four counters, respectively. In each version of the
     problem, the number of spaces on the board was always one more than the number
     of counters. For the 2-, 3- and 4-counter problems students were provided with the
     boards and counters so that they could work on the problem together.

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PART 2: Insights about mathematics learning: Collaborative problem solving

The pairs who successfully solved the 2-, 3- and 4-counter problems were invited to    Prompts that could be used while students worked on a problem:
work out how many moves would be required if the problem was extended to 10
and then 1000 counters. The students were not provided with additional counters or        •   Maybe start by telling your partner what you need to do.
boards for these extended problems. Instead a pen and recording sheet were made               If there’s very little talking at the start, or if the students seem unsure how to begin.
available. There were no rules about how materials should be used or shared.              •   Remember to talk with your partner about what you’re thinking.
Students were allowed a maximum of 15 minutes to work on the task. While students             Anytime, to help elicit their thinking.
were working, the TA could use the following prompts to support the students. The
                                                                                          •   Why are you stuck there?
TAs were expected to use their professional judgement to decide when these were
needed.                                                                                       When students stop because they have two same-colour counters in a row,
                                                                                              blocking other moves.
                                                                                          •   Is there a different move you could make at this point?
                                                                                              Especially if students repeatedly get stuck at the same point and might be
                                                                                              becoming frustrated.
                                                                                          •   Just stop for a moment and think about what will happen if you move there.
                                                                                              What are the other moves you could make?
                                                                                              Especially if students are getting to the same move and getting blocked by having
                                                                                              two counters of the same colour together.

                                                                                      TAs could also say to the students: ‘You can move more than one of your counters
                                                                                      in the same turn.’ Many students expected to take turnabout, probably because
                                                                                      moving one counter at a time is often the convention when counters are used
                                                                                      in board games. However, as this picture illustrates, students quickly reached an
                                                                                      impasse by turn-taking.

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PART 2: Insights about mathematics learning: Collaborative problem solving

     Solutions                                                                    Assessing the task
     The figure below shows the number of moves needed for each version           We developed scoring guides (rubrics) to assess each pair’s problem-solving and
     of the problem as recorded by a pair of Year 8 students.                     collaboration skills. While a problem-solving rubric was developed for each of the
                                                                                  problems, a single scoring guide was developed for collaboration. For both problem
                                                                                  solving and collaboration, students were assessed as a pair rather than individually.
                                                                                  The rubric for collaboration was applied at two different stages of the task. The first
                                                                                  stage incorporated the work students did up to and including the 4-counter problem.
                                                                                  The second stage involved the work done on the 10-counter and 1000-counter
                                                                                  problems. Trained markers were used to score the task .
                                                                                  The scoring guide for collaboration (see following page) focused on the social
                                                                                  interactions between the students as they looked to solve the problems. Evidence
                                                                                  of collaboration included the way the students communicated with each other
                                                                                  (including the use of body language such as gestures, facial expressions and
                                                                                  exclamations), shared resources and information, and took on various roles.

     A completed table for the Moving and Jumping task

32
PART 2: Insights about mathematics learning: Collaborative problem solving

Scoring guide for collaboration

   Score
            Collaboration criteria
 category
     2      • Students interact with each other throughout the task to work out
              what they need to do to solve the problem together.
            • Students share the work.
            • Students consistently show in their actions that they are listening
              to their partner’s ideas.

     1      • Students interact with each other occasionally during the task to
              work out what they need to do to solve the problem together.
            • One student leads and the other student is also engaged in the
              task.
            • Students sometimes show in their actions that they are listening
              to their partner’s ideas.

     0      • There is no / almost no interaction about what students need to
              do to solve the problem together.
            • One student leads / dominates and the other is largely passive.
            • Students never / almost never show in their actions that they are
              listening to their partner’s ideas.

                                                                                    33
PART 2: Insights about mathematics learning: Collaborative problem solving

     Insight 1:
     Most students can collaborate on a task that is engaging and clearly structured

     Most students at both year levels showed that they could collaborate in a clearly                  Student A                                 Student B
     structured and engaging problem-solving context. In the first stage of the Moving
     and Jumping task nearly all the pairs at each year level scored a ‘1’ or more on the
     collaboration rubric. Forty-seven percent of the Year 4 pairs and 66 percent of the Year 8
     pairs scored a ‘2’ (the top category). The pairs of students who scored in the top category   Maybe you could move that
     were able to sustain their collaboration. They shared resources and information and           [points to a blue counter],
     actively listened to each other.                                                                        there.
     Collaboration depends on motivation and a sense of challenge. The task provided a                                            [She follows his suggestion]
     sense of challenge. It was presented as a problem activity and invited curiosity and                                                 Move that one
     experimentation. Students could have multiple attempts at the problem and finding                                              [points to a red counter],
     a solution was satisfying. The task was also structured so that no one student could                                                     there.
     access all the resources. This meant each student in a pair had to be at least minimally
                                                                                                     You could move that one
     involved.
                                                                                                   [points to a blue counter],
                                                                                                              there.
                                                                                                                                            Right
                                                                                                                                    [moves the counter]
                                                                                                                                 then you can move that one
                                                                                                                                   [points to a counter],
                                                                                                                                            there.
                                                                                                   [He moves a counter.]

                                                                                                                                   And then I’ll move this one .
     The following interaction from a pair of Year 4 students is typical of the kind of
     communication associated with a score of 2 for collaboration during the first stage of
     the task. The pair of students has just succeeded in solving the 3-counter problem and
     are trying to repeat their solution and count the number of moves that were required.                                        [moves a counter.]
     Each student is prepared to give and listen to suggestions.

34
PART 2: Insights about mathematics learning: Collaborative problem solving

Insight 2:
Working systematically and making links between problems enhances task success

Most pairs of students were able to successfully solve the 2-counter version               As this pair of Year 4 students sets up the 3-counter problem and begins moving the
of the problem using a trial and error approach (sometimes with support).                  counters, there are already clues that they are on the way to successfully completing
Students who were successful with the 2-counter problem were often able to                 the problem.
successfully complete the 3- and 4-counter problems. The results below show the
scoring guide we used for the 2-counter problem and how the students achieved.

Results for Moving and Jumping, by year level                                Y4 Y8               Student A                                                  Student B
                                                                              88    87
                                                                             pairs pairs
  Students’ responses to the 2-counter problem, for pairs of students         n     n

     Score                                      Problem-solving criteria
                                                                                                        I know how we can do it,
  3                                                   Correct answer (8).                              just do the same thing but
		                                      Students solve the problem and                                          with more.
		                                       can repeat it with relative ease. 30 60
  2                        Correct answer reached, maybe with support.
		                                     Students solve the problem once                                                              It’s just the first one was hard.
		                                      and may repeat it with difficulty. 49 25                                                       Now we know what to do.

  1                                         Students begin the problem
		                                             but are unable to solve it.    7     2
  0                                          No response / Don’t know /
		                                  Unable to begin solving the problem       2     0

                                                                                                                                                                                   35
PART 2: Insights about mathematics learning: Collaborative problem solving

     Students were more likely to progress through the series of problems when they
     worked systematically and used what they had learned about the activity along the
     way. For instance, the pairs who took notice of the patterns that emerged when they
     applied trial and error on the 2-counter version were more successful at solving the 3-
     and 4-counter versions. The figure shows the number of pairs who began each of the
     five different problems by year level. Students who did not begin all the problems were
     either not able to solve a problem and therefore could not move on or ran out of time.
     In general, Year 8 students began more problems than those in Year 4. Eighty-eight
     pairs of Year 4 students and 87 pairs of Year 8 students began the first task.

                                                                                               The number of pairs of students who began each problem in the Moving and Jumping task,
                                                                                               by year level

36
PART 2: Insights about mathematics learning: Collaborative problem solving

Insight 3:
Thinking algebraically involves using a range of problem-solving strategies

Although many of the student pairs did not get beyond the 4-counter problem in the
15 minutes provided, 13 pairs of Year 4 students and 41 pairs of Year 8 students did go
on to complete the 10-counter problem. The table shows how the students scored on
the 10-counter problem.

Results for Moving and Jumping, by year level                                                                                                                           Y4 Y8
  Students’ responses to the 10-counter problem, for pairs of students, by year level                                                                                   13    41
                                                                                                                                                                       pairs pairs
      Score                                                                                                                               Problem-solving criteria      %     %
         3                                                                                                                                    Correct answer (120)
		                                                                                                          • Students solve the problem, using algebraic thinking
		                                                 • Students identify a relationship between the number of counters and number of moves needed to swap them,
		                              and express a rule for the number of moves needed using algebraic notation, e.g., n(n + 2) where n is the number of counters each.       -    12
  2		                                                                                                                  Correct answer reached, maybe with support:
			                                                                                                                                      • Students solve the problem.
		                         • Students take a systematic approach to solving the problem. From the first 3 problems, they identify a relationship between the number
		                                          of counters and number of moves needed to swap them and describe a rule for predicting the number of moves needed
		                                                • Students may use a sequential approach to solve the problem, e.g., identify the difference between 8 and 15 (+7),
		                                                                                                                   and 15 and 24 (+9), and then add 11, 13, 15, etc. 15     16
  1		                                                                                                • Students are unable to solve the problem or solve it with difficulty
			                                         • Students treat this as a stand-alone problem, i.e., they do not make links to the easier problems to identify a relationship
			                                                                                                      between the number of counters and number of moves needed
		                      OR      • Students make links to the easier problems but mis-hypothesise a relationship between the number of counters and moves needed 54            60
 0		                                                                                                                           • No response / Don’t know / Guessed
			                                                                                        • Students’ combined explanations are vague, disorganised, or incomplete
			                                                                                                         • Students do not pinpoint what made the problem hard
			                                                                                                           • Students begin the problem but are unable to solve it 31      12

                                                                                                                                                                                     37
PART 2: Insights about mathematics learning: Collaborative problem solving

     For the 10-counter problem and the 1000-counter problem that followed it, students          The pairs of students who did solve the 10-counter task generally identified a
     were no longer able to rely on using physical materials and applying a series of moves.     sequential (or recursive) pattern that predicted the number of moves needed as
     Instead, they had to be able to think algebraically in order to identify a pattern that     the number of counters in the problem increased. They made the table the focus of
     linked the numbers together. Students who succeeded on these problems often                 their inquiry. They often recorded additional information in the table to support their
     demonstrated that they could think algebraically. These students could apply a range        thinking.
     of strategies to search for patterns that provided a more generalised solution. Students    One pair of students worked out the number of moves that would be needed for
     who relied on a concrete approach struggled. For instance, one student decided that         a 1-counter problem (3 moves). They then focused on how the number of moves
     drawing the 10-counter problem would be helpful.                                            increased as more counters were added to the problem. They recorded the following,
                                                                                                 noticing that the number of moves required was increasing by consecutive odd
                                                                                                 numbers.

     When this didn’t enable them to solve the problem, their partner suggested that they
     could just use ‘pretend counters’ on the table. However, even with prompting from
     the TA, the difficulties involved in working with larger numbers and the lack of
     concrete materials to ‘anchor’ the moves proved too hard to overcome.

           Algebra in Moving and Jumping
           These problems give students an opportunity to show the extent to which they
           can identify and express rules for patterns. In NZC, this particularly links to the
           bolded text in the following achievement objectives in Patterns and relationships:
              •   Level 1: Create and continue sequential patterns
              •   Level 2: Find rules for the next member in a sequential pattern
              •   Level 3: Connect members of sequential patterns with their ordinal
                  position and use tables, graphs, and diagrams to find relationships
                  between successive elements of number and spatial patterns
              •   Level 4: Use graphs, tables and rules to describe linear relationships
                  found in number and spatial patterns.
           To support students to focus on thinking algebraically, a simple table was
           provided for each pair to record the number of moves each problem took.

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