Unpacking Student
Responses for
Teacher Information
Workshop presented at
National Numeracy Facilitators Conference
February 2008
Teresa Maguire, Jonathan Fisher and Alex Neill
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Outline
Supporting teachers with the ARBs (10 min)
Resources with fractions (20 min)
Resources with algebra (20 min)
Other information on resources (25 min)
Discussion (15 min)
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Supporting teachers with the ARBs
How the ARBs can be used to support
teachers?
Concept maps
Animation / CD
Next steps booklet
Support material
Teacher information pages
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Concept maps
Provide information about the key
mathematical ideas involved
Link to relevant ARB resources
Suggest some ideas on the
teaching and assessing of that
area of mathematics
Are “Living” documents
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Concept maps
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Animation CD/exe
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Next steps booklet
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Support materials
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Teacher information pages
Task administration
Answers/responses
Calibration
easy (60-79.9%)
Diagnostic and formative information
(common wrong answers and misconceptions)
Strategies
Next steps
Links to other resources/information and to concept maps
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Eating fractions of pie, pizza and cake
(NM1251)
The questions
Student responses and misconceptions
Strategies
Suggested Next steps
Other resources
Discussion
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Eating fractions of pie, pizza and cake
(NM1251)
167 students
Year 6
Nationwide
Range of
deciles
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Eating fractions of pie, pizza and cake
(NM1251)
Petra ate two-fifths (2/5) of a pizza
and Sarah ate one-fifth (1/5).
Show how to work out how much
pizza they ate altogether.
3/5
(64%, and 66% showed working)
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Eating fractions of pie, pizza and cake
(NM1251)
Lima and Paul each had the same
sized cake. Lima ate four-fifths (4/5) of
his cake and Paul ate three-fifths (3/5)
of his cake. Show how to work out how
much cake they ate altogether.
7/5 or 1 2/5
(38%, and 52% showed working)
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Eating fractions of pie, pizza and cake
(NM1251)
Bill ate one-fifth (1/5) of a whole
apple pie. Show how to work out
how much pie was left.
4/5
(64%, and 70% showed working)
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Eating fractions of pie, pizza and cake
(NM1251)
Andrew started with one and a half
pizzas (1 1/2) and ate three-quarters
(3/4) of a whole pizza. Show how to
work out how much pizza is left.
3/4
(46% - 53% showed working)
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Eating fractions of pie, pizza and cake
(Misconceptions)
Numbering the Pieces only
3
7
4
3
Whole number (top and bottom)
3/10
7/10
Other whole number relationship/system
13 17 or 4 8 or 4/10 8/10 or 4/0
8/0
Varying referent whole
1/2 or 3/6
And the “Size of the pieces”
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Eating fractions of pie, pizza and cake
(Next steps)
Partitioning
Part-whole relationships
Referent whole
Whole class discussion
Explanation and justification
Diagrams
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Eating fractions of pie, pizza and cake
(Other resources)
Link to other ARB resource
(keywords)
Fractional thinking concept map
NEMP
Book 7: Teaching Fractions,
Decimals and Percentages, 2006
Figure it out
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Balance pans & Solving simple equations
AL 7111 & AL7124
=
www.nzcer.org.nz/arb
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Strategies with Zero
Student strategies
a)
i) 25 + 16 – 16 =
ii) 28 + 36 – 36 + 52 – 52 =
iii) 62 + 74 – 62 =
iv) 78 – 44 + 44 =
v) 67 + 23 + 55 – 23 – 67=
b)
Explain what you did
Usage and success rates using the additive identity concept
In our sample, 37% of students included at least something in part b) to indicate
that they were employing the concept of the additive identity. These
accounted for 43% of the strategies students described.
- 13% of students had an explanation including reference to the concept of the
additive identity and no calculations were used. These students got an
average of 94% of their answers in part a) correct, and maintained their
success throughout the five parts of the question.
- 19% applied the additive identity rather than a calculation but were unable to
clearly explain what they had done. 85% of these students' answers in
part a) were correct, but they were slightly less successful with the harder
questions.
- 5% had an explanation that included reference to the additive identity but
they also did some calculations. Only 67% of these students' answers in
part a) were correct, but they were less successful with the harder
questions.
The remaining 63% of students described computational methods, other
methods, or gave no explanation in part b). These accounted for 57% of
the strategies students described. These students had lower success
rates that varied from 0% to 68%, and their success rates generally
dropped off on the harder questions.
For more details on the success rates of student strategies, click on the link Appendix of
student strategies for AL7123
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Strategies with Zero
Description of strategy in part b)
Number of
students
Percent of
strategies1
Percent of
students2
24
15%
13%
94%
35
22%
19%
85%
9
6%
5%
67%
Computation strategy
1.Calculates left to right
2.Uses vertical algorithm
3."Used my fingers"
4.Other calculation
5.Incorrect calculation (e.g., ignores
some numbers or operators)
47
8
2
23
4
29%
5%
1%
14%
2%
25%
4%
1%
12%
2%
49%
68%
50%
32%
0%
Other explanations
10
6%
5%
54%
TOTAL strategies used
162
100%
86%
62%
No explanation
26
14%
27%
TOTAL
188
100%
57%
Additive identity strategies
1.Explanation includes reference to
the additive identity and no
calculations used.
2.Applies the additive identity rather
than a calculation but unable to
clearly explain what they had done.
3.Explanation includes reference to
the additive identity but some
calculations are also included.
100%
Success
rate3
Based on a representative sample of 188 students
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Car maintenance & Post a parcel
AL6156 & AL 7129
Exemplars of student responses
- Graphs
Multiple representations
- Graphs, tables, and equations
Other goodies
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Car maintenance
(AL6156 – Level 4)
b) Draw a line graph
Incorrect line graphs:
Another type of graph:
Step graph
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Diagnostic and formative information
Common error
Likely misconception
a) i)
ii)
110, 150, 190, 230, 270, 310
145, 205, 265, 325, 385, 445
Confuses intercept (cost for a first look) and
slope (hourly cost).
Adds on $40 or $60 for each hour of
servicing.
a) i)
ii)
140, 210, 280, 350, 420, 490
170, 265, 350,435, 520, 605
Ignores the intercept (40 or 60) and assumes
the total charge for 1 hour equals the
slope.
b)
Incorrect use of line graphs:

joins origin to end point;

bend the line to pass through the origin;

starting at 1 hour.
Ignores or misinterprets the role of the
intercepts ($40 or $70), preferring an
intercept of $0.
b)
Uses an other type of graph:

Histogram

Bar graph (often a series of vertical lines)

Scatterplot

Relationship graph
c)
Makes no comparison between the two garages.
e.g., "It's cheaper to use the garage at the start because the longer you're in
the garage, the more expensive it is."
c)
Honore's, because it has a lower hourly cost.
Example: "Honore's Garage is cheapest because for every hour it's only $25."
Ignores the effect of the lower intercept (set
up cost of $40) for Honore's garage.
c)
Kakariki's because it is cheapest at the beginning of the table.
Ignores the effect of the lower slope (hourly
rate of $25) for Honore's garage.
c)
Identifies Honare's or Kakariki's as cheapest with no justification.
Based on a representative sample of 201 students.
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Car maintenance – Student Responses
joins origin to end point
bend the line
starting at 1 hour
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Histogram
Bar graph
Relationship graph
Scatterplot
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Car Maintenance
Step graph
This graph could be seen as correct if only complete hours are charged
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Post a parcel
(AL7129 – Level 5)
Multiple representations
.
Graphs
Tables
Equations
Two courier companies
a)
This graph shows the
price Peru.
Show how to use this
graph to find the weight
of a parcel that both
companies would charge
the same
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Post a parcel-Student strategies
Graphical
interpretation [part a)]
.
Indicating the x- and the y- coordinates of the intersection of the lines (5%).
Indicating the x-coordinate only of the intersection of the lines (35%).
Indicating the intersection of the lines only (26%).
Using graphical interpolation to get a more accurate answer (19%). This was often used in
conjunction with one of the other three strategies. About 80% of students who showed
interpolation obtained a correct answer, while only about 50% of those whose working did not
show interpolation got a correct answer. Many of the latter were satisfied with 2.5 as their
answer, even though the break-even point was clearly somewhat less than that.
Table interpretation [part b)]
Interpolation of the table between 1 and 2 kg (4%).
Using the differences between 1 and 2 kg for the two companies (9%).
Averaging the costs at 1 and 2 kg (1%).
Using equations [part c)]
Algebraic formulation of the problem (9%).
Trial and improvement methods (15%). The included a range of methods including moving
sequentially towards the solution in single integers, to jumping several numbers to speed up
getting to the answer.
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Graphical interpretaton
Indicating the intersection of the lines only
Indicating the x-coordinate only
Indicating the x- and the y- coordinates
Graphical interpolation
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Table interpretation
Interpolation of the table
Weight of parcel (in kilograms)
Company
0
1
2
3
4
FastAir
$20
$60
$100
$140
$180
$220
SafeWay
$50
$70
$90
$110
$130
$150
Using the differences between 1 and 2 kg
70 - 60 = 10
100 - 90 = 10
5
Averaging
(60 + 100) / 2 = 80
(70 + 90) / 2 = 80
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Using equations - Algebraic formulation
FastAir:
Cost = 7x + 14
SafeWay:
Cost = 4x + 35
Formulation only
7x + 14 = 4x + 35
Formulation and attempt to solve
Formulation and a successful solution
7x + 14 = 4x + 35
7x = 4x + 49
3x = 49
x = 16.33
7x + 14 = 4x + 35
3x = 21
x=7
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Using equations – Trial and improvement
Initial guess only (sometimes correct)
Initial guess then iterating with a step
size of 1
Initial guess then other guesses
Jumping towards the solution.
Starting from 1 and iterating in steps of 1
7 x 1 + 14 = 21
4 x 1 + 35 = 39
7 x 2 + 14 = 28
4 x 2 + 35 = 43
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Future areas
Statistics
Geometry and Measurement
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Enlargement
(GM5118)
Terminology of
enlargement
Scale factor??
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Next steps
Uses an additive model
Students need to look carefully at the language clues in the question. The phrase "how many times bigger …
compared with" in parts b)ii) and c)ii) needs to be interpreted as a multiplicative question. If students have
misinterpreted the question this way, see if they can perform the question once this has been clarified.
Unfamiliar with the term "scale factor" or of the concept of enlargement
Work with students to understand that enlargement increases each dimension linearly by an amount known as
the scale factor. Get the students working with enlargement on grid paper. GM5013 describes the process
without using the term scale factor. Click on Level 3 and 4, enlargement AND scale factor for further resources
that assess this. Also click on the Figure it out resource Enlargement Explosion (Geometry, L4+, Book 2, page
12).
Unfamiliar with the effects of enlargement on area or volume
Students firstly need to know that "scale factor" is a linear measure. They then need to explore the relationship
between scale factor and area.
The increase in area goes up by the square of the scale factor. This is because area is a square measure (m2,
cm2 etc). Click on scale factors AND area OR invariant properties for further resources on this relationship.
Resources GM5113, GM5038, and GM5054 are particularly useful for teaching this principle. Also click on the
Figure it out resource Growing Changes (Geometry, L3, page 24).
The increase in volume goes up by the cube of the scale factor. This is because area is a cubic measure (m3,
cm3 etc).
Click on scale factors AND volume for further resources on this relationship. Resources GM5113, and GM5038
are particularly useful for teaching this principle.
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Perimeter
(MS2178)
Composite shapes
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Next steps
The first four diagnostics above or other incorrect answers indicate that the student does not know how to calculate the
circumference of a circle. These students need to have experiences in how to calculate the circumference. ARB resource
MS2107 and Figure it out resource Circle Links (Measurement, L4, Book 1, page 3) each give an activity that gets
students physically measuring the circumferences and diameters of different circles and exploring the relationship between
them.
The classical definition of π is geometrical, and it is the ratio between the circumference and the diameter of any
circle.
π = circumference ÷ diameter
(of any circle)
So a circle with a diameter of 1 unit has a circumference of π units.
Students could also be asked to explore π (pi) on the internet.
The following are some historical fractional approximations to pi:
Numerical value
Fractional form
Date and Origin
3
3/1
2000 BC Biblical
3.125
25/8
2000 BC Babylonian
3.16049
(16/9)2
2000 BC Egyptian
3.14285
22/7
250 BC Archimedes
3.14167
377/120
150 AD Ptolemy
3.14159292
355/113
480 AD Chung Chi
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008
Assessment Resource Banks
www.arb.nzcer.org.nz
Username: arb
Password: guide
Teresa Maguire, Jonathan Fisher, Alex Neill
February 2008