Download Mathematics Standard Grade (2006 and 1998)

Comparison of Mathematics
Standard Grade 2006 and 1998
Summary
Syllabus
Although the content of the Standard Grade courses has become more
extensive, there is no overall change in the demand in either the breadth or
depth of the syllabus.
Question papers
There has been a change to the structure of the examination, but there are no
significant differences between 1998 and 2006 in the level of demand of
either the Knowledge and Understanding or the Reasoning and Enquiry
elements in the examination questions.
Marking Instructions
Improvements to the clarity of Marking Instructions have provided greater
guidance in partial marking in 2006. This has proved beneficial to the
marking process. It has not altered the overall demands of KU and RE for
each grade.
Scripts
On the basis of the small sample of scripts at each grade for 1998 and 2006,
no overall differences were found in the candidate evidence at any grade.
1
1
Findings
The panel’s remit was to compare the 1998 and 2006 Mathematics Standard
grade courses, the course contents, the component elements, the assessment
arrangements, and the marking procedures, to determine the relative
demands of syllabus and assessment in the two years. This was carried out
by studying the changes in the course content, the coverage of the syllabus
in the examinations, the structure and demands of the examinations, and the
candidate responses as shown by a given sample of scripts. There were
separate papers for each of Foundation, General and Credit levels; most
candidates take the examination at more than one level.
In 1998, 99.5% of SG candidates were from S4 and in 2006 it was 96.3%. A
small increase (from 1% in 1998 to 3% in 2006) of S3 pupils sitting
standard grade has been noted. So the populations in 1998 and 2006 were
very similar.
There have been several changes in the structure of the course. In 1998 there
were three Elements, KU (Knowledge and Understanding), RA (Reasoning
and Applications), and an Investigation, and these were weighted in the ratio
2:2:1. By 2006 several changes had taken place at each level.
♦ The Investigation had been removed and the two other Elements, now
called KU and RE (Reasoning and Enquiry), were equally weighted. RE
had precisely the same description as RA.
♦ Some new Statistics material had been introduced.
♦ A non-calculator paper had been introduced.
The introduction of a non-calculator paper has led to many schools
introducing non-calculator lesson starters throughout the year; this fits in
well with the non-calculator element in 5-14.
Several other developments have helped in the preparation of pupils for the
examinations: marking schemes are now made available on the web by
SQA; there has been an increase in the number of revision books available,
and many centres now provide revision sessions outwith school hours.
The move from a 2:2:1 to a 1:1 weighting has had beneficial results for
some candidates. For example, in 2006, a candidate achieving a 1 in KU and
a 2 in RE, gains an overall 1 due to the better of the two grades being
awarded. In 1998, when the Investigation contributed to the final grade, a
rounding up of half a grade never occurred. However, in 1998 the
investigation grades tended to be higher than the KU and RA grades (see
Appendix 6), so perhaps some compensation has occurred.
2
2
Level of demand of arrangements/syllabus
2.1
Demands of breadth
Although the content of the courses has become more extensive (due mainly
to the introduction of statistics and the non-calculator content), there is no
evidence that the 2006 course is more demanding than 1998 in KU or RE.
The only change in the Elements examined was that the Investigation
Element had been removed by 2006. The reasoning aspect of Investigations
has to some extent been assimilated into the reasoning element RE.
There have been several changes in the Arrangements:
♦ In 1998 the Arrangements required that Foundation candidates were
trained to answer oral questions; this skill had been removed by 2006.
♦ By 2006 there was a significant emphasis on non-calculator skills. This
was achieved by introducing a separate non-calculator paper at each of
the three levels.
♦ There have been some minor changes in the syllabuses in addition to the
introduction of Statistics. These changes are described in Appendix 1.
2.2
Differences in depth
Although there is, in 2006, more detail in the exemplification notes in the
contents, there is no overall change in the depth demanded in either KU or
RE.
The introduction of a non-calculator paper has enabled non-calculator skills
to be integrated with a number of KU skills, such as understanding of
percentages, indices, brackets and substitution into a formula. Apart from
this, there was no evidence of different emphases on integration of skills
between 1998 and 2006.
In conclusion, there was no evidence of any significant change in the
breadth or depth of the overall demands in KU or RE from 1998 to 2006.
3
Level of demand of examination questions
3.1
General approach
The 1998 and 2006 assessment questions were fairly similar in style, but
there was evidence of some movement from context-based questions to
context-free questions, particularly in the KU element of the opening
questions of the non-calculator papers.
3
3.2
Coverage of the syllabus
It is inevitable, given the constraints of time, that only a sample of grade
related criteria can be tested in any one year. Some topics can be expected to
appear on a regular basis, but others will be examined only from time to
time. The coverage for 1998 and 2006 is shown in Appendix 2. In Appendix
3 we give the mark distributions between KU and RE in the two papers; in
Foundation and General the distribution is fairly balanced in each, but in
Credit the non-calculator paper has proportionally more KU marks, while
the second paper has more RE marks.
3.3
Structure
In 1998 the investigation contributed 20% of the total assessment, and, at
each level, KU and RA were examined in one paper. In 2006 there were two
papers at each level. The lengths of these papers are given in Appendix 4,
where it can be seen that the length of the papers has decreased at
Foundation level but has remained unchanged at General and Credit. The
availability of more marks in the Credit papers has enabled more partial
marks to be awarded. Further, the splitting up of the exam into two papers
not only gives candidates a break but also provides the opportunity of a
‘fresh start’ with easier questions at the start of the second paper, and this is
probably to their advantage.
3.4
Demands of assessment tasks/questions
There is evidence of a move towards breaking up longer questions into
stages. For example in the 1998 Credit paper there was a 6-mark question
with no internal subdivision, whereas in 2006 there was no question or
subpart worth more than 4 marks. Indeed in 2006 there was a subpart of the
form ‘show that...’ which made the marks of the next part more readily
available. Overall, more partial marks in both KU and RE were accessible in
the 2006 paper.
In conclusion, although the structure of the examination has changed, there
are no significant differences between 1998 and 2006 in the level of demand
of the examination questions in both KU and RE.
4
Level of demand of examination marking
Markers in 2006 were given ‘illustrations of evidence’ (rather than an
annotated descriptive response) and further notes giving greater guidance in
partial marking, to allow for greater consistency between markers. Many
possible candidate responses are listed, with the corresponding marks to be
awarded. These improvements in the marking instructions do not change the
overall demands of KU and RE for each grade, but provide greater
clarification.
4
In the sample of 1998 scripts provided for the panel, there were examples of
candidates who perhaps would have received higher marks if more partial
marks had been annotated in the marking instructions. This is particularly
true of the 1998 General paper, where this problem was perhaps
compensated for by the low grade boundaries for RA at levels 3 and 4 (see
Appendix 5).
5
Grading of candidates’ performances
The panel was given a small sample of 1998 and 2006 scripts at each grade.
Due to the archive selection process used in 1998, the scripts were chosen
near the middle of the grades whereas by 2006 the selection process had
changed and scripts were chosen near the grade boundaries.
5.1
Comparability of performance
On the basis of this small sample, no overall significant differences were
found at any level between the candidates’ performances in the two years.
5.2
Strengths and weaknesses
♦ The introduction of the first paper enabled the 2006 candidates to
demonstrate their non-calculator skills.
♦ The panel looked at the sample papers at different levels to compare
performances. On the basis of this small sample there was evidence in
the Grade 2 scripts of a better performance in trigonometry in 1998, but
an improvement of algebraic skills in 2006. But it is dangerous to read
too much into this as the contexts in which these skills were tested were
different.
♦ In two of the three scripts obtaining a Credit 2, the RE performance in
the non-calculator paper was noticeably poor.
The panel concluded that, overall, scripts with the same grades for KU and
RE in 1998 and 2006 show comparable candidate performance.
5
Appendix 1: Content and syllabus changes
Changes in Knowledge and Understanding – Extended GRC
K15 use simple statistics
Foundation
♦ Calculate mean and mode of a data set;
♦ Calculate mode of data presented in an ungrouped frequency table.
General
♦ Calculate median and range of a data set;
♦ Calculate mean, median and range of data presented in an ungrouped
frequency table;
♦ Draw a best-fitting line by eye on a scattergraph and use it to estimate
the value of one variable given the other;
♦ State the probability of a simple outcome.
♦ Interpret calculated statistics e.g. compare mean and range for two sets
of data.
Credit
♦ Calculate the quartiles and semi-interquartile range from a data set or
ungrouped frequency table;
♦ Calculate, given the formula, the standard deviation of a data set;
♦ Determine the equation of a best-fitting straight line (drawn by eye) on a
scattergraph and use it to estimate a y-value given the x-value;
♦ Find probability defined as:
— number of favourable outcomes total number of outcomes where all
outcomes are equally likely.
Changes in Reasoning and Enquiry – Extended GRC
R16 Reason and draw valid conclusions where appropriate
Credit
♦ Assess the significance of the results.
6
Content Checklist
Changes from 1998 to 2006
Deletions at Foundation Level
♦ Ratio
♦ Angle –Sum of angles of a triangle
♦ 3-figure bearings – measuring the bearing of B from A
Additions at Foundation Level
♦ Distance, speed, time - finding distance given speed and time, simple
cases only.
♦ 3-figure bearings – from a diagram read off bearing e.g. radar screen.
♦ Generalising number pattern – simple rule in words.
♦ Graphs and Tables:
— Extracting data from pictograms, bar charts, line graphs, pie charts and
scattergraphs.
— Constructing pictograms, bar charts, scattergraphs and line graphs from
given data.
— Constructing a frequency table from data without class intervals.
♦ Use of simple statistics:
— Calculating mean and mode of a data set.
— Calculating mode of data presented in an ungrouped frequency table.
— Interpreting calculated statistics.
Deletions at General Level
♦ Constructing Triangles
Additions at General Level
♦ Calculating time intervals – Change decimal time into actual time.
♦ Solving triangles – Right angled triangles using sine, cosine and tangent
(excluding cases where the unknown in the ratio is in the denominator).
♦ Graphs and Tables:
— Interpreting data from stem-and-leaf diagrams (charts)
— Constructing stem-and-leaf diagrams (charts)
— Drawing a best-fitting line by eye on a scattergraph and using it to
estimate the value of one variable given the other.
♦ Use of simple statistics:
— Calculating median and range of a data set.
— Calculating mean, median and range of data presented in an ungrouped
frequency table.
— Stating the probability of an outcome.
7
Deletions at Credit Level
♦ Properties of Circles - tangents from an external point.
♦ Relationships among sine, cosine and tangent functions – for
complementary angles, e.g. sin 70 0 = cos 20 0 ;
♦ For angles greater than 90 0 , relation to acute angle, e.g. cos 200 0 = -cos
20 0
♦ Linear programming
Additions at Credit level
♦ 3-figure bearings – calculating the bearing of A from B, given the
bearing of B from A.
♦ Graphs and Tables
— Extracting data from boxplot and dotplot.
— Constructing a pie chart, boxplot and dotplot.
— Constructing a cumulative frequency column for an ungrouped
frequency table.
— Calculating the quartiles and semi-interquartile range from a data set or
ungrouped frequency table.
— Calculating, given the formula, the standard deviation of a data set.
— Determining the equation of a best-fitting straight line (drawn by eye) on
a scattergraph and using it to estimate a y-value given the x-value.
— Knowing that probability is a measure of chance between 0 and 1.
Non-calculator numerical skills
These have been introduced at each level, involving whole numbers,
integers, decimals, fractions and percentages. For detailed description see
pages 52 and 53 of the Standard Grade Arrangements in Mathematics,
published by SQA, April 2000.
8
Appendix 2 Coverage of syllabus in assessments
Credit
Topic
scientific notation (significant figures)
graphs & tables
volume of composite solid
extending number pattern & generalising
find the equation of a line
similar triangles
trigonometry (right-angled triangle with
unknown denominator)
arc length
perpendicular bisector of chord (and Theorem
of Pythagoras)
extract data from table
construct formula to describe relationship
construct formula to describe relationship
cosine rule (unknown side)
simultaneous equations
change the subject of a formula
solve a quadratic equation
sine rule (& right-angled triangle )
Variation
extending number pattern
construct formula to describe relationship
solve a trigonometric equation
solve an equation with algebraic fractions
factorise a quadratic expression
add / subtract algebraic fractions
substitute into a trigonometric function
maximum / minimum values of a
trigonometrical function
simplification of surds
use of index laws
order of operations
add / subtract fractions
substitute into a function
simplify algebraic fractions
solve a linear equation (with brackets)
similar volumes
area of triangle (1/2 ab sinC)
area of triangle (1/2 bh) - height unknown
solve an inequality
calculate circumference (significant figs)
calculate mean & standard deviation
interpret calculated statistics
undo a percentage increase / decrease
multiply out an expression
Theorem of Pythagoras & right-angled
trigonometry (unknown angle)
3 figure bearings
volume of a prism
substitute into a formula
question
1
2
3
4
5
6
1998
marks
2
3
4
5
3
3
element
KU
RA
KU
RA
KU
KU
7a
3
RA
7b
4
RA
8
4
KU
9a
9b
2
4
RA
KU
10
11a
11b
11c
12
13
14a
14b
15
16
17a
17b
18a
3
3
2
3
6
4
1
3
3
3
2
2
3
KU
KU
KU
KU
RA
KU
RA
RA
KU
RA
KU
KU
RA
18b
3
19a
19b
2
2
9
paper
2006
question
marks
element
1
2
4
11
3
4
KU
RE
2
8
4
RE
1
1
2
1
2
2
8
11a,b
6b
9
7b
9c
2
1, 3
4
1, 5
4
3
RE
KU/RE
RE
KU/RE
RE
RE
1
5a
1
KU
2
10a
2
KU
RA
2
10b
4
RE
KU
KU
2
2
1
1
1
1
1
1
1
1
1
2
2
2
2
2
4c
4b
1
2
3
5b
6
7
10a
10b
11
1
2a
2b
3
4a , 9b
2
2
2
2
2
2
3
4
1
3
3
2
3
2
3
1,2
KU
KU
KU
KU
KU
KU
KU
KU
KU
RE
RE
KU
KU
RE
KU
KU
2
5
4
KU
2
2
2
6a
7a
9a
2
1
2
RE
KU
KU
General
Topic
Add / subtract integers (in context)
money calculations (multiply , subtract)
express one quantity as a percentage of
another
enumerate possibilities
find a fraction of an amount
express one quantity as a fraction of another
exchange rates
solve an inequality
time intervals
rotational symmetry
extract data from tables
money calculations (multiply, subtract, divide)
communicate information in graphical form
form a linear equation
solve a linear equation
use Theorem of Pythagoras
find area of a right-angled triangle
find dimensions of cuboid given volume and
other dimensions
factorise an expression
money calculations (rounding, add)
money calculations (divide, multiply)
find the gradient of a line
draw a line with given equation
find circumference
appropriate rounding
extending number patterns
find a formula to generalise a pattern
substitute into a formula
find unknown using proportion
use trigonometry to find unknown angle / side
in right-angled triangle
add / subtract decimals
multiply divide by multiples of 10, 100, 1000
find percentage of an amount
find price of reduced item (fractions)
convert to scientific notation
probability
extract data from a line graph
construct stem-and-leaf diagram
calculate the median
find unknown angles (radii, tangent, triangles)
money calculation (overtime)
distance, speed and time calculation
calculate area of circle
money calculation (time intervals)
scale drawing (bearings)
money calculation (fraction then percentage of
a given amount)
question
1
2a
1998
marks
2
2
element
KU
KU
2b
2
KU
3
4a
4b
5a, b
6
7
8
9a
4
1
2
5
2
1, 4
3
1
RA
KU
KU
KU
KU
KU/RA
RA
KU
9b, 9c
9d
10a
10b
11a
11b
5
2
2
2
4
2
KU
RA
RA
RA
KU
KU
12a, b
5
RA
13
14a
14b
15a
15b
16a
16b
17a
17b
17c
18
2
2
3
2
2
3
1
2
2
4
2
KU
KU
RA
KU
KU
KU
RA
RA
RA
RA
KU
19
4
RA
10
paper
2006
question
marks
element
2
3
4
RE
2
2
1
10a
3
2
KU
KU
2
2
6b
8
3
4
KU
KU
2
6a
2
KU
1
5
2
KU
1
1
1
2
4a
4b
4c
2
2
2
3
3
RE
RE
RE
KU
2
4
4
KU
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
1a
1b, c
1d
2
3
6
7
8a
8b
9
5
7
9a, b
10b
11
1
2
2
3
2
1, 2
4
3
1
3
4
4
2, 2
3
3
KU
KU
KU
KU
KU
KU/RE
RE
KU
KU
RE
RE
RE
KU/RE
RE
RE
2
12
4
RE
Foundation
Time
1998
Question
1
Marks
3KU
2006
Question
6(P1)
Marks
4KU
Scale
2
3KU
Money
3
4KU
Time
4
4RA
Symmetry
5
4RA
Symmetry
2 (P2)
4RE
Supple. Angles
6
2KU
Angle
type
measure
4 (P1)
2KU
Volume
7
4RA
Volume
14(P2)
4RE
Plan
8
1KU,3RA
Bar Chart
9
4KU
Bar Chart
7(P1)
1KU,3RE
Combinations
10
1KU,3RA
Combinations
9(P2)
3RE
%
increase
11
2KU,3RA
%
discount
10(P2)
4KU
Measure
12
3KU
Number Pattern
13
6RA
Number Pattern
5(P2)
6RA
Average
average
rounding
14
5KU
Mode, Mean
6(P2)
5KU
Scale
15
1KU,3RA
Scale
4(P2)
1KU,3RE
Time
16
3KU
Time
3a(P2)
1KU
Table
17
2KU,3RA
ml distribution
18
3RA
Weight distribution
5(P1)
2RE
Area of Triangle
19
3KU
Area of Triangle
7(P2)
3KU
Money
20
3KU,3RA
Money
11(P2)
5KU
Topic
Topic
Time
11
Appendix 3: Distribution of marks
1998
Foundation*
General
Credit
KU
RA
KU
RA
KU
RA
2006
40
39
42
41
42
40
paper 1
(non-calc)
13
11
17
16
22
16
KU
RE
KU
RE
KU
RE
paper 2
(calculator)
27
29
23
24
23
29
total
40
40
40
40
45
45
* Note: Foundation 1998 includes 10 KU and 4 RA marks from four oral questions.
Appendix 4: Duration of exams (in minutes)
1998
Foundation
General
Credit
70
90
135
2006
paper 1
(noncalc)
20
35
55
paper 2
(calculator)
total
40
55
80
60
90
135
Appendix 5: Grade Boundary comparison
1998
KU
1
2
3
4
5
6
RA
1
2
3
4
5
6
2006
%
34/42 = 81%
22/42 = 52%
31/42 = 74%
21/42 = 50%
26/40 = 65%
18/40 = 45%
1998
%
31/40 = 78%
23/40 = 58%
24/41 = 58%
15/41 = 37%
27/39 = 69%
18/39 = 46%
KU
1
2
3
4
5
6
RE
1
2
3
4
5
6
12
%
36/45 = 80%
26/45 = 58%
30/40 = 75%
21/40 = 53%
27/40 = 68%
19/40 = 48%
2006
%
29/45 = 64%
18/45 = 40%
31/40 = 78%
23/40 = 58%
25/40 = 63%
18/40 = 45%
Appendix 6: Grades attained
Overall
1
2
3
4
5
6
7
999
Overall
1
2
3
4
5
6
7
999
Overall
1
2
3
4
5
6
7
999
%
18.82
13.44
22.37
16.98
19.55
7.28
1.46
0.10
%
16.0
14.0
21.0
17.0
21.0
6.0
1.0
2.0
%
2.82
-0.56
1.37
-0.02
-1.45
1.28
0.46
-1.90
2006 Mathematics
KU
%
RE
1
18.36
1
2
15.40
2
3
19.42
3
4
16.79
4
5
19.54
5
6
8.18
6
7
2.26
7
800
0.04
800
%
13.81
11.76
20.13
20.51
19.11
10.63
3.98
0.07
1998 Mathematics
KU
%
RA
1
13.2
1
2
15.7
2
3
16.9
3
4
20.8
4
5
21.9
5
6
7.8
6
7
3.8
7
%
11.1
13.3
18.8
21.6
16.7
13.1
5.4
change from 98 to 06
KU
%
RA/RE
1
5.16
1
2
-0.30
2
3
2.52
3
4
-4.01
4
5
-2.36
5
6
0.38
6
7
-1.54
7
%
2.71
-1.54
1.33
-1.09
2.41
-2.47
-1.42
13
Practical
1
2
3
4
5
6
7
%
31.4
11.6
26.5
10.5
16.3
2.6
1.1