Impact of a distance mathematics course

Description and impact of a distance mathematics course for
grade 10 to 12 teachers

Gerrit Stolsa, Alwyn Olivierb and Diane Graysonc

a,cUniversity of Pretoria and bUniversity of Stellenbosch

Email: agerrit.stols@up.ac.za, baio@sun.ac.za and c dgrayson@absamail.co.za

This paper explores the impact of a one-year in-service distance education mathematics course that
follows a problem-solving approach. The course aims to develop teachers’ mathematical thinking skills.
Results show that the course improved the teachers’ confidence, problem-solving skills and teaching
skills. Results also show that the course on mathematical thinking skills helps to improve teachers’
understanding of mathematics content.

Introduction and background
The fact that there is a crisis in mathematics
education is common knowledge and a cause for
concern. For the past five years only approximately
20,000 grade 12 learners (4%) a year have passed
mathematics on Higher Grade (HG) in the Senior
Certificate examinations (Department of Educ-
ation, 2001; Kahn, 2004). Given that a pass in HG
mathematics is a prerequisite for entry into
science-based studies at university, there are
serious implications for the country’s ability to
produce enough scientifically skilled professionals.
It was with these points in mind that a distance-
based course for grade 10-12 teachers was
developed by the Centre for the Improvement of
Mathematics, Science and Technology Education
(CIMSTE) at the University of South Africa
(UNISA). The course was designed and written by
the second author in 2001 and 2002 and
implemented by the first author for the first time in
2002. The focus of the course is on problem-
solving, in the context of algebra.

In the planning phase of a course, an important
question is what the course should look like in
order to have the maximum impact possible on
teachers and their learners. What kind of mathe-
matical skills and knowledge do teachers need in
order to teach effectively? It is impossible to
answer such questions before we have decided
what we want from the learners. We naturally want
the learners to be able to do mathematics, but then
the question arises: What is mathematics and how
do you learn it? It is therefore important, when
designing a course, to start with a view of the
nature of mathematics and the learning of
mathematics. Dossey, McCrone, Giordano and
Weir (2002) believe that there is no general
agreement on the question:

The wide variety of its [mathematics’]
applications in our society is easy to list.
But, the nature of mathematics itself is
hard to capture. This results from a lack
of consensus, even among
mathematicians, as to what constitutes
‘mathematics’ and what ‘doing mathe-
matics’ means. (2002: 4)
In reality, most professional mathematicians

spend little thought on the fundamental nature of
their subject. What we want to know is: which
skills do we value as mathematical skills? This can
change according to changes in our society.
Confrey and Lachance explain why:

These skills (computational skills)
allowed students to secure jobs and to
become informed citizens in an industrial
society. However, with advances in
technology, such computational skills are
no longer as important. Instead, students
need to develop critical-thinking skills to
interpret data appropriately and to use
technology to solve more complex
problems. Thus, changes in our society
have led to a change in what we value in
mathematical skills. (2000: 232)
We currently support the following view

expressed in the South African National
Curriculum Statement on grade 10-12 mathe-
matics (Department of Education, 2003):

Mathematics enables creative and logical
reasoning about problems in the physical
and social world and in the context of
mathematics itself. It is a distinctly
human activity practised by all cultures.
Knowledge in the mathematical sciences
is constructed through the establishment
of descriptive, numerical and symbolic
relationships. Mathematics is based on

32 Pythagoras 65, June, 2007, pp. 32-38

Gerrit Stols, Alwyn Olivier, and Diane Grayson

observing patterns which with rigorous
logical thinking, leads to theories of ab-
stract relations. Mathematical problem-
solving enables us to understand the
world and make use of that under-
standing in our daily lives. Mathematics
is developed and contested over time by
social interaction through both language
and symbols. (2003:7)

In terms of this statement, we want the learners to:
• Solve problems, using creative and logical

reasoning.
• Find relationships and express them

symbolically.
• Observe patterns to find relationships and prove

them.

To expand on this, in the words of the Education
Development Centre (2000), “Students need to be
thinkers: pattern hunters, experimenters,
describers, tinkerers, inventors, visualisers,
conjecturers, guessers and seekers of reasoned
argument and proof.” We can describe justifying,
explaining, analysing, generalising and defining as
mathematical thinking. We believe the best way to
help learners to develop the ways of thinking that
are characteristic of mathematics is through
problem-solving (Education Development Centre,
2000: xiv). According to Resnick (1989), theory
and research show that we develop habits and
skills of interpretation and meaning construction
though a process of socialisation or enculturation
rather than through instruction:

…becoming a good mathematical prob-
lem-solver – becoming a good thinker in
any domain – may be as much a matter of
acquiring the habits and dispositions of
interpretation and sense-making as of
acquiring any particular set of skills,
strategies, or knowledge. If this is so, we
may do well to conceive of mathematics
education less as an instructional process
(in the traditional sense of teaching
specific, well-defined skills or items of
knowledge), than as a socialisation
process. In this conception, people
develop points of view and behaviour
patterns associated with gender roles,
ethnic and familial cultures, and other
socially defined traits. When we describe
the processes by which children are
socialised into these patterns of thought,
affect, and action, we describe long-term
patterns of interaction and engagement in
a social environment. (1989: 58)

This view of enculturation highlights the
importance of perspective and point of view as core
aspects of knowledge. The case can be made that a
fundamental component of thinking mathematic-
ally is having a mathematical point of view, or
having a mathematical attitude of mind – seeing
the world in the way mathematicians do.

The focus of our course is therefore on
developing a mathematical attitude of mind and the
way to do it is to immerse participants in a typical
mathematical culture.

Theoretical context
One of the most important factors influencing
learner performance is the teacher. In a synthesis
of research related to the President’s Education
Initiative, Taylor and Vinjevold (1999) indicate
that there are problems with teachers’ knowledge
and skills and, in consequence, with their teaching
approaches:

… reform initiatives aimed at revitalising
teacher education and classroom
practices must not only create a new
ideological orientation consonant with
the goals of the new South Africa. They
also need to get to grips with what is
likely to be a far more intractable
problem: the massive upgrading and
scaffolding of teachers’ conceptual
knowledge and skills.
…the fundamental mechanism for its
propagation [the vicious cycle of rote
learning] is the lack of conceptual
knowledge, reading skills and spirit of
enquiry amongst teachers. (1999: 160)

What is needed if mathematics teachers are to
become more effective are professional develop-
ment opportunities to strengthen their conceptual
knowledge and problem-solving skills. It comes as
no surprise that a focus of the new South African
Further Education and Training (FET) curriculum
is on helping learners develop problem-solving
skills because problem-solving is central to the
constructivist-based teaching of mathematics. A
problem-centred learning approach is based on the
acceptance that learners construct their own
mathematical knowledge. The difficulty that arises,
however, is that the teachers are not trained to
teach problem-solving and did not experience the
power of a problem-centred teaching approach
themselves. The way in which teachers have been
taught themselves plays an important role in the
way they think about teaching. It is therefore
important in a course for the training of teachers to

Description and impact of a distance mathematics course for grade 10 to 12 teachers

allow those teachers to experience problem-solving
first hand. In a report, the National Commission on
Teaching and America's Future (1996:20) discuss
the fact that teacher preparation and professional
development programmes must consciously
examine the expectations embodied in new
curriculum frame-works. The report also
comments on the need for these programmes to
develop strategies that help teachers learn to teach
in these much more demanding ways. Teachers’
programmes, according to Loucks-Horsely,
Hewson, Love, and Stiles (1998: 36), must be
organised around problem-solving and must be
directly related to teachers' work with their
students.

This paper discusses the impact of a problem-
solving course on a group of 27 teachers, in terms
of both their mathematical understanding and their
attitudes towards mathematics and the teaching of
mathematics.

Description of the course
The purpose of the UNISA (University of South
Africa) Mathematics for Teachers course, for
which the students receive 12 credits (an average
teacher will take about 120 hours to complete the
course successfully), was to improve the teachers’
mathematical thinking skills and pedagogical content
knowledge, mainly in the context of problems. The
duration of the course is one year and is offered by
means of distance teaching. UNISA is a distance
education institution, a factor which poses
challenges in itself. To try to overcome some of the
problems presented by a distance course (e.g.
sharing solutions through discussions with others),
we held three workshops of two hours each. We
encouraged the teachers to form peer groups by
explaining the advantages of such groups to them
and sending them a list of the telephone numbers
and addresses of all the students enrolled for the
course.

The central focus of the course is on problem-
solving, that is, non-routine problem-solving –
either through illustrations of the process of
problem-solving, or through students’ own
engagement with problems. In order to encourage
the teachers to reflect regularly on the mathematics
that they had learnt and to think of ways to
introduce their new-found knowledge in the
classroom, we asked them to keep and submit a
journal and encouraged them to update their
journals regularly – at least once a week. As part of
the course, teachers were required to complete four
assignments, which we marked and returned to
them.

Examples of problems from the study guide
The examples below are taken from the course
study guide. They will give the reader a better idea
of what we mean when we ask learners to find
relationships and express them symbolically and
observe patterns to find relationships and prove
them.

Example 1: Short cut
a) Develop a short method to calculate

17 + 16 + 15 + 14 + … + 3 + 2 + 1.
b) Use your method to calculate

1 + 2 + 3 + 4 + 5 + … + 99 + 100.
c) Generalise!
d) Can you use your method to calculate

2 + 4 + 6 + 8 + 10 + … + 98 + 100?
e) Can you use your method to calculate

1 + 3 + 5 + 7 + 9 + … + 97 + 99?

Example 2: Consecutive numbers
Some numbers can be written as the sum of two or
more consecutive whole numbers.

For example: 13 = 6 + 7
14 = 2 + 3 + 4 + 5

15 = 7 + 8 = 1 + 2 + 3 + 4 + 5

Some numbers cannot be written as the sum of
consecutive whole numbers.
Investigate: Which numbers can and which
numbers cannot be written as the sum of
consecutive whole numbers. Try to develop a
general theory or method or formula that will
enable you to:
a) Immediately decide if any given number can be

written as the sum of consecutive numbers.
b) Easily write the number as the sum of

consecutive numbers.

Example 3: Generalise
a) Find the value of

(1 –
1
4 )(1 –

1
9 )(1 –

1
16 )(1 –

1
25 )(1 –

1
36 ) … (1 –

1
10000 )

b) Generalise.

Example 4: Petrol price
In January the petrol price is increased by 10%.
Then, in February the petrol price was reduced by
10%. John says that the petrol price is now the
same as it was before the first increase. Is this
correct? Explain!

Impact of the course on teachers
In 2002, twenty-seven teachers of grades 10 to 12
mathematics enrolled for the Mathematics for
Teachers course. The majority of the teachers were

Gerrit Stols, Alwyn Olivier, and Diane Grayson

teaching at rural black schools. The reflective
journals provided us with documentation of a
continuous cycle of inquiry. Apart from the
journal, various data were gathered in order to
assess the impact of the course. At both the
beginning and the end of the course the teachers
completed a questionnaire and a test. The
questionnaire asked teachers to indicate their level
of confidence about teaching various topics in the
grades 10, 11 and 12 syllabi using a 5-point Likert
scale. The test comprised questions similar to those
on the grade 12 examinations and included
questions with a range of levels of cognitive
demand (see Appendix A). The same questionnaire
and test were administered at both the beginning
and the end of the course. Further information was
obtained from teachers’ assignments, journals,
evaluations completed at the end of workshops and
an end-of-year evaluation.

Effects on teachers’ content knowledge
At the end of the course we asked the teachers to
evaluate the course by completing a questionnaire.
We divided the free response questions into
categories. Thirteen teachers completed and
returned the questionnaire/survey. The respondents
gave feedback on the impact of the course on their
content knowledge and teaching practice as
follows:
1) Has the course improved your subject content

knowledge?
Yes: 13 (100%).
No: 0.

2) In what ways has the course improved your
subject content knowledge?
Improved problem-solving skills: 6 (46%).
Improved mathematical thinking skills: 3

(23%).
No response: 4 (31%).

3) In what ways has the course failed to improve
your content knowledge?
None: 11 (85%).

4) Has the course improved your teaching
practice?
Yes: 13 (100%).
No: 0.

5) In what ways has the course improved your
teaching practice?
More confidence: 1 (8%).
Learners are more interested: 1 (8%).
Improved learners’ class attendance: 1 (8%).
Improved teachers’ problem-solving skills: 3

(23%).
Improved teaching skills: 3 (23%).

The pre- and post-tests showed that there was a

13.6% improvement in the teachers’ ability to
answer grade 12 exam-type Higher Grade
questions (see Figure 1). The average mark was
32.4% for the pre-test and 46% for the post-test.

The improvement in the teachers’ ability to
answer grade 12 exam-type Higher Grade
questions was unexpected, because the course did
not explicitly focus on the content knowledge that
was tested in the pre- and post-test.

Some of the comments that the teachers made
in the journals were:

I am discovering something new every
day in my learning. My knowledge
horizon is expanding although I have
been a teacher for more that 15 years.
I have discovered ways of attempting a
problem if you don’t have a clue of what
to do. It has helped me to upgrade my
mathematical insight.

Effects on classroom practice
Anecdotally, the course had an impact on the
classroom practice of the teachers. Some of the
comments they made were:

The weeks’ work will affect my classroom
practice in that I will simply be moving
away from the traditional way of
teaching. I will make sure that the
learners learn through social interaction
and reflection, no longer through
practice and repetition. I will present
tasks and problems that will lead the
learners to inventing mathematics.
This piece of work will enrich the
knowledge of my learners as far as
arithmetic sequence is concerned. This is
very exciting, as they will have a deeper
understanding of the concept.
The learners are more interested in
mathematics, understand it more quickly
and attend their classes better.
All of these exercises empower you to
teach certain topics… it is true that
experience is the best teacher.

Effects on teachers’ attitudes towards
mathematics
From the feedback it was evident that all the
respondents believed that the course had improved
their teaching practice. According to them they are
more confident than before about teaching
mathematics. One of the teachers wrote:

One is empowered to teach in the
classroom without fear. The students

Description and impact of a distance mathematics course for grade 10 to 12 teachers

understand me more clearly
than before I registered for
this course.

The course also changed the way
that some of the teachers think
about mathematics:

It was very challenging and
has changed my mindset
about mathematics so that I
now realise that mathematics
is about discovery and about
being able to make con-
jectures and to reason.

Effects on teachers’ reflective skills
Initially the teachers were very negative about the
weekly journal entries, but their attitude changed
as they progressed. In the beginning one of the
teachers wrote: “I think that the journal exhausts
our study time”. Later in the year, the same teacher
wrote:

I am now developing a positive attitude
about this journal. I can see that the
questions in the journals help us to link
what we have learnt in the study guide
with the activities in the classroom
situation.
Twenty of the 27 teachers were from the same

geographical area, which contributed to the fact
that 77% of the teachers worked with peers. The
journals and the fact of working with peers helped
the teachers to be more reflective. The journal gave
the researchers some insight into what was going
on in the teachers’ minds during the problem-
solving course. In the beginning of the course the
teachers expressed their frustrations:

This was too demanding and thought
provoking. One had to recognise patterns
which were not always easy to find.
The assignment is really challenging. I
didn’t expect this. I am thinking of
stopping this course.
As I am working through this unit, I am
getting frustrated … I see the problems
for the first time.
Some of the teachers complained about the

study guide. They wanted examples, followed by
similar problems to solve. They looked for
examples in other books:

As I was working through this unit, I was
frustrated. I was frustrated to see myself
reading so many books and not finding
the exact book that would provide me
with relevant information.

But the frustration soon changed into joy. The
same teacher wrote some time later:

My frustration ended in excitement. Even
though I was frustrated, at the end of
each and every question I had gained
something exciting.

One teacher wrote in her journal: “I am really
frustrated. I wish I could get the answer.” Two
days later she wrote:

At last I got the answer and I am so
happy. No amount of money can buy my
happiness.

Possible impact of the course on the
learners
The study only determined the impact of the course
on the teachers. Although we attended some of the
teachers’ classes, we could not determine the
impact of the course on their classroom practice.
Instead of trying to determine the direct impact of
the course on the learners, we decided to divide the
investigation into two parts. The first question we
asked was: is there a correlation between the
teachers’ knowledge and their learners’ know-
ledge? The second question was whether this
course improved the teachers’ knowledge.

What we know is that there is a strong
correlation between the teachers’ knowledge and
their learners’ knowledge. A previous study
undertaken by Stols (2003: 246-250) with the same
teachers revealed that the correlation between these
teachers’ ability to answer exam-type questions
and the learners’ ability to answer similar exam-
type questions is very strong (the Pearson
correlation coefficient is 0.8008; this correlation is
statistically significant, because of the low P-value
of 0.0005).

Therefore, improving the teachers’ ability to
answer exam-type questions will also improve
their learners’ ability (see Figure 2). In the results,
we mentioned that the teachers’ ability to answer

Mathematical
thinking course

Teachers’ ability to
answer exam-type

questions

Improved by 13,6%

Figure 1. Improvement in the teachers’ ability to answer

exam-type questions

Gerrit Stols, Alwyn Olivier, and Diane Grayson

exam-type questions improved by 13,6%. It is
therefore reasonable to believe that this problem-
solving course in mathematical thinking could help
the learners as well. This belief was confirmed by
the results for one school. The pass rate of the
grade 12 learners in Mathematics Higher Grade in
this school was 51.4% in 2001 and it improved to
94.4% in 2002 (that is the year in which the
mathematics teacher at the school was enrolled for
this course). The teacher believes that the
improvement in the results was due his particip-
ation in this course.

Discussion
The course improved the confidence, problem-
solving skills and teaching skills of the teachers.
One of the teachers who did the course wrote in his
journal:

I believe that creativity is there in each
and every mind. It only needs to be
activated in order to make it useful in
other situations.
The course also improved the teachers’ content

knowledge because it helped them to help
themselves to master content. It is clear from the
comments by the teachers that this course
(problem-solving approach) helped the teachers to
improve their metacognitive skills. A possible
reason is that teachers have become more
reflective because of the journals they kept and
another possible reason is that a problem-solving
approach may help teachers to enhance their
metacognitive skills. Routine exercises on the
other hand may engender a false impression of
success and understanding because it is possible to
experience success without understanding.

We would conclude that teacher training
programmes that are organised around problem-
solving and that are directly related to teachers’
work with their students are a powerful means of
helping teachers who are preparing to teach
mathematics. This approach will help teachers to

become more reflective and will develop their
metacognitive skills. It will help them to help
themselves in future.

Conclusion
To prepare teachers to help learners learn
mathematics is not easy. Traditionally, the focus of
teacher training programmes was on the upgrading
of content knowledge and on ways of explaining
the new knowledge to learners. But Schoenfeld
(1994) states that teachers need more than that:
“The danger in this kind of ‘content inventory’
point of view comes from what it leaves out: The
critically important point is that mathematical
thinking consists of a lot more than knowing facts,
theorems, techniques, etc.” It remains important,
however, that the teachers should at least know the
mathematics they teach. Yet, considering the
requirements outlined above, it is clear that they
must know more than that. Ball (2003) explains the
knowledge that teachers need as follows:

Teaching requires justifying, explaining,
analysing errors, generalising, and
defining. It requires knowing ideas and
procedures in detail, and knowing them
well enough to represent and explain
them skilfully in more than one way.
This is mathematics. The failure to
appreciate that this is substantial
mathematical work does teachers – and
the improvement of teaching – a dis-
service. (2003: 4)

This course helps teachers deepen their content
knowledge and their pedagogical content
knowledge, improve their problem-solving skills,
and develop their metacognitive skills so that they
can continue to learn in future without relying on a
structured course. Because of the strong correlation
between teachers’ knowledge and learners’
knowledge, the course will eventually make a
difference in the classroom.

Teachers INSET Course Learners

Relationship?

Correlation: 0,8008 (P-value: 0,0005)

Figure 2. The relationship between teachers’ and learners’ abilities

Description and impact of a distance mathematics course for grade 10 to 12 teachers

mathematical problem-solving (pp. 32-60).
Reston, VA: National Council of teachers of
Mathematics.

Acknowledgements
This paper was made possible by a grant from the
Carnegie Corporation of New York. The
statements made and views expressed are solely
those of the author.

Schoenfeld, A.H. (1994). What do we know about
mathematics curricula? Journal of Mathematics
Behaviour, 13, 55-80. Stols, G.H. (2003). The correlation between
teachers’ and their learners’ mathematical
knowledge in rural schools. In P. Bongile, M.
Dlamini, B. Dlamini, & V. Kelly. (Eds.),
Proceedings of the 11th Annual SAARMSTE
Conference (pp. 246-250). Cape Town:
University of Cape Town.

References
Ball, D.L. (2003). Mathematics in the 21st

Century: What mathematical knowledge is
needed for teaching mathematics? Retrieved
November 28, 2003, from http://www.ed.gov/
inits/mathscience/ball.html

Confrey, J. & Lachance, A. (2000). Transformative
teaching experiments through conjecture-driven
research design. In A.E. Kelly. (Ed.), Handbook
of research design in mathematics and science
education (pp. 231-266). New Jersey: Lawrence
Erlbaum Associates.

Taylor, N. & Vinjevold, P. (1999). Getting
Learning Right. Johannesburg: Joint Education
Trust.

Department of Education. (2001). National
strategy for mathematics, science and
technology education to address the problem in
mathematics, science and technology education.
Pretoria: Department of Education.

Appendix A: Pre- and post-test
Please answer the following questions without
obtaining help from anyone.

1 ;
1

6
≠≤

−
xx

x
1. Solve for x:

2. For which values of k are the roots of

k
12

2
2 =++

+
xx

x
with x ≠ –1 real?

Department of Education. (2003). National
Curriculum Statements Grades 10-12
(General): Mathematics. Pretoria: Department
of Education. 3. Solve for x: 9

x + 3x = 27(3x + 1)
4. A house contractor has subdivided a farm

into 100 building lots. He has designed two
types of homes for these lots: colonial and
ranch style. A colonial home requires
R300 000 of capital and yields a profit of
R40 000 when sold. A ranch-style house
requires R400 000 of capital and yields an
R80 000 profit. If he has R36 million of
capital on hand, how many of each type
should he build for maximum profit? Will
any of the lots be vacant?

Dossey, J., McCrone, S., Giordano, F. & Weir,
M.D. (2002). Mathematics methods and
modelling for today's mathematics classroom: a
contemporary approach to teaching grades 7-
12. Canada: Brooks & Cole.

Education Development Centre (2000). Connected
geometry: Teachers guide. Chicago: Everyday
Learning.

Kahn, M. (2004). For whom the second bell tolls:
Disparities in performance in Senior Certificate
Mathematics and Physical Science. Perspec-
tives in Education, 30(4), 149-156. 5. If ax2 + bx + c = 0 and a + b + c = 0 find the

numerical value of x. Loucks-Horsely, S., Hewson, P., Love, N. & Stiles,
K. (1998). Designing professional development
for teachers of science and mathematics.
Thousand Oaks, CA: Corwin Press.

6. Show that a = 0 if
ba2ba2

ba2aba2a2
2

−
−−+

=
xx

National Commission on Teaching and America's
Future (1996). What Matters Most: Teaching
For America's Future. Summary Report. New
York: National Commission on Teaching and
America's Future.

7. Solve for x and y: ( ) 0142 =⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
−−

y
yx

8. Is the following statement true:

( ) 122121 −− +=+ aaaa ? Why? Olivier, A. (2000). Study Guide: Mathematics for
Teachers I. Pretoria: UNISA Publishers. 9. Is the following statement true:

2(4x + 4–x) = 8x + 8–x ? Why? Resnick, L. (1989). Treating mathematics as an ill-
structured discipline. In R. Charles & E. Silver
(Eds.), The teaching and assessing of 10. Solve for x: (x

2 + 2x)2 − x2 − 2x − 6 = 0

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/JPEG2000ColorImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 30
>>
/AntiAliasGrayImages false
/DownsampleGrayImages true
/GrayImageDownsampleType /Bicubic
/GrayImageResolution 300
/GrayImageDepth -1
/GrayImageDownsampleThreshold 1.50000
/EncodeGrayImages true
/GrayImageFilter /DCTEncode
/AutoFilterGrayImages true
/GrayImageAutoFilterStrategy /JPEG
/GrayACSImageDict <<
/QFactor 0.15
/HSamples [1 1 1 1] /VSamples [1 1 1 1]
>>
/GrayImageDict <<
/QFactor 0.15
/HSamples [1 1 1 1] /VSamples [1 1 1 1]
>>
/JPEG2000GrayACSImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 30
>>
/JPEG2000GrayImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 30
>>
/AntiAliasMonoImages false
/DownsampleMonoImages true
/MonoImageDownsampleType /Bicubic
/MonoImageResolution 1200
/MonoImageDepth -1
/MonoImageDownsampleThreshold 1.50000
/EncodeMonoImages true
/MonoImageFilter /CCITTFaxEncode
/MonoImageDict <<
/K -1
>>
/AllowPSXObjects false
/PDFX1aCheck false
/PDFX3Check false
/PDFXCompliantPDFOnly false
/PDFXNoTrimBoxError true
/PDFXTrimBoxToMediaBoxOffset [
0.00000
0.00000
0.00000
0.00000
]
/PDFXSetBleedBoxToMediaBox true
/PDFXBleedBoxToTrimBoxOffset [
0.00000
0.00000
0.00000
0.00000
]
/PDFXOutputIntentProfile ()
/PDFXOutputCondition ()
/PDFXRegistryName (http://www.color.org)
/PDFXTrapped /Unknown

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>>
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice