May 2023 subject report
Mathematics: applications and interpretation SL TZ2
Standard level internal assessment
The range and suitability of the work submitted
A greater variety of topics were seen this session. However, as in previous sessions many of the
explorations were of the correlation type using PMCC, 𝜒 2 test for independence or goodness of fit test.
There were more explorations using the two-sample t-test in this session. A greater percentage of students
also applied modelling and calculus with varying degrees of success. A few students attempted
mathematics above or outside the SL syllabus. These were seldom successful as the students were unable
to show evidence of understanding the chosen processes.
There were still explorations seen where the students would give an introduction, some contrived reason
for interest, a plan that is sometimes not justified at the beginning and not always adhered to, followed
by a mechanical application of textbook type processes irrespective of the relevance in context. This was
most evident in the statistics-based explorations where the students failed to provide a thorough
description of the sampling technique and process and neglected to justify the sample size. The students
often failed to establish the relevance to the aim of many of the processes applied. Most of these
explorations included comments about the need for more data without considering the strengths and
limitations of their initial sampling technique and whether this resulted in a representative sample of the
population.
A greater number of modelling type explorations were seen. Some more successful than others. Students
either found the models manually or used technology, either approach being acceptable.
There seemed to be less explorations using the application of Voronoi diagrams. Some explorations on
the probability of winning in games such as Monopoly and Backgammon were noted. There were few
explorations seen on topics such as Geometry and Finance.
It was noted that in some schools all student
each student looked at different variables to compare which limited the scope for personal engagement
as well as reflection.
Most of the work submitted used mathematics commensurate with the course, there were however
exceptions that either included mathematics beyond or entirely from the prior knowledge sections: these
did not score well overall.
Student performance against each criterion
Criterion A
An overall improvement was noticed in that students now place lengthy tables of data and/or calculations
in the appendix and as well as refer and comment on graphs/diagrams/tables when introduced.
Unfortunately, there still are some persistent issues such as: an unclear aim/title or an aim which is slightly
different from the title which was confusing (to the student and the reader); including processes irrelevant
to the aim just for the sake of hav
more mathematic
This plan also sometimes presumed outcomes that were yet to be determined which affected coherence.
In some explorations there were many titles and subtitles without linking the sections through
explanations and reasons for performing the processes. This affected coherence as well as organization.
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© International Baccalaureate Organization 2023
May 2023 subject report
Mathematics: applications and interpretation SL TZ2
In some instances, the students neglected to check if the conclusion was consistent with the aim which
affected the coherence greatly.
Students should be made aware that long, often textbook type, explanations of mathematical processes
in the syllabus are unnecessary and could affect coherence, organization as well as conciseness.
It should be noted that placing many scatterplots/graphs/calculations one after the other and only then
discussing the results can affect the coherence as well as the organization.
Criterion B
Students should be able to achieve high levels in this criterion. However, through lack of careful proof
reading and incorrect use of mathematical notation and mathematical terminology students are not
achieving the desired levels.
To have a variety of representations, students often include inappropriate and/or irrelevant diagrams and
graphs such as pie charts, bar graphs and histograms. In the further explanations section of Criterion B in
the Teacher Support Material (TSM)
Level B4 can be achieved using only one form of
It should be noted that mathematical terms such as: correlation , proof , accurate , significant and
test all have specific definitions and meaning in mathematics and should be used appropriately.
Unless the formulae for the PMCC, the linear regression line, t-test and the 𝜒 2 -test for independence
and/or Goodness of Fit are used to calculate these values, they should not be presented in the exploration.
When the graphic display calculator (GDC) is used to find these values, the formulae are considered
irrelevant.
The 𝜒 2 -test for independence and/or GOF and the t-test each have a specific format which should be
adhered to. Many students used the incorrect format and as well as symbols and names for these tests.
Null and alternate hypotheses should only be used in context of statistical tests and not be used as an
attempt to indicate personal engagement.
When initially graphing a scatterplot to investigate any trends, students should ensure that the regression
line is not drawn. Regression lines on a scatterplot are only appropriate when these are referred to and are
relevant to the aim.
Variables should always be defined in context of the exploration and the aim. Variables must be italicized
in text as well as in the mathematical processes. Students must be made aware that interchanging upperand lower-case letters indicating the same variable is inappropriate.
The use of computer notation such as *, ^, E etc is still in evidence but seems to be less than in previous
sessions.
Formatting should be checked to ensure that equations/formulae/expressions are not broken and run
over two lines and are centred on the line. It is not appropriate to show trivial mathematical calculations
such as finding the mean, median, mode and quartile values, all of which can be determined using
technology.
Calculator instructions and/or screenshots of all the steps taken on said calculator is not considered to be
appropriate presentation and should not be included.
Students must avoid explaining mathematical processes in paragraph form and should show these steps
in mathematical notation form only.
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© International Baccalaureate Organization 2023
May 2023 subject report
Mathematics: applications and interpretation SL TZ2
Many graphs are still presented without axes labels, badly noted values on the axes, incorrect domain and
range in context of the data.
It was also observed that students use 𝑦 = or 𝑓(𝑥) notation interchangeably. When 𝑓(𝑥) is used then
the use of
𝑑𝑦
𝑑𝑥
is inappropriate. If more than one model function is found then each should be labelled
differently, all cannot be noted as 𝑦 = or 𝑓(𝑥) = .
Criterion C
Effort is certainly a part of this criterion, but this is not all that is needed. Collecting primary data (some
without any proof of this in evidence) but then continuing to complete textbook maths is not an indication
of significant personal engagement.
Stating an interest in the topic, sometimes quite contrived, is not required and does not provide evidence
of personal engagement.
Learning new maths is one way of indicating personal engagement. However, the maths should be
appropriate and relevant to the aim. Applying more complex processes when there are easier ways to
accomplish the same outcome is unfortunately not an indication of personal engagement.
Explorations based purely on finding the correlation and/or the independence between two variables
rarely score well in this criterion.
Some students noted personal interest in a topic, sometimes with long discussions and explanations
which was then followed by no engagement in the mathematical exploration of these ideas. Stating
interest and a summary of research is not an indication of personal engagement in the development of
the exploration.
Students using a template like approach were more prevalent this session. These types of explorations
seldom allowed students to achieve C2/3.
There were, however, more students who attempted to use different approaches, perspectives, and
strategies in achieving their aim which allowed the awarding of higher levels in this criterion.
Criterion D
Students should pause and reflect on their work at each stage of the development.
Unfortunately, reflection is still seen as something to be done at the end in the paragraph entitled
Stating personal opinions not substantiated by the results cannot be considered as critical or even
meaningful reflection.
Writing a summary of the processes and results found in the development of the exploration as well as
some personal opinions in the conclusion section cannot be considered as meaningful reflection.
It should be noted that meaningful reflection on only one aspect of the exploration and providing limited
reflection on the results and techniques used in the rest of the exploration is not sufficient evidence of
meaningful reflection overall. Example: meaningful reflection on why certain values are outliers in a data
set; then not stating a conclusion regarding the inclusion/exclusion of these values in further calculations;
providing limited reflection on the results of the statistical processes which follow.
Meaningful reflection could be seen in the following:
• choice of approach
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© International Baccalaureate Organization 2023
May 2023 subject report
Mathematics: applications and interpretation SL TZ2
• limitations of approach in context
• effect that assumptions could have on the validity of the results
• reasonability of results in context of the exploration.
Reflection should be used to consider alternative approaches, the mathematical processes and used to
create questions which could drive the exploration forward. Merely stating possible strengths and/or
extensions to the explorations does not validate meaningful reflection.
Criterion E
In this session, most of the explorations contained maths commensurate with the level of the course.
However, students and teachers should be made aware that including maths from the syllabus does not
always guarantee E3. If the student is unable to demonstrate even limited understanding or if the maths
were not relevant to the aim/topic then the highest level that can be awarded is E2.
Students continue to find it difficult to explain/justify the use of processes in context of the aim such as
t-tests.
Students are still confused between causation/effect/affect/correlation in the interpretation and
application of the PMCC value.
In the explorations focused on statistical methods, students tended to apply statistical processes without
considering whether the nature of the data collected satisfied the assumptions/conditions of the statistical
processes chosen. It was noted that students often spend half the exploration on calculating univariate
the relationship between two variables without explaining
the relevance of the univariate stats and often not using any of the results (these explorations feel very
template-like, lacking understanding and engagement).
Students seldom gave
rank correlation coefficients and mostly
interpreted the results incorrectly. Finding no or weak correlation, students then continued their
exploration by completing the 𝜒 2 -test for independence and/or GOF as well as the student t-test when
the awarding of levels in this criterion.
In the modelling type explorations students often selected random points with no justification to
determine the parameters of the chosen model function using simultaneous equations. It would be more
appropriate to use technology and regression to find the chosen model function. Most frequently used
models were polynomial and exponential models. Top students commented on the shape of the data and
justified their choice of model by clearly explaining the assumptions made as well as the characteristics of
the phenomenon being modelled in context of the aim and considered the appropriate domain for the
model. In successful modelling explorations students also considered and commented on the fit of the
model to the collected data.
find a mathematical model to fit data or merely state which model will be used without really looking at
how appropriate the models were in context of the exploration. Using only the 𝑅 2 (coefficient of
determination) to determine how valid the model is, is an insufficient reason for choice of model and
evidence of limited understanding of the modelling process.
Completing maths outside the syllabus, such as topics from HL, need to be explained in such a manner
that the reader (who should be assumed to have general SL subject knowledge) will understand it.
Textbook style explanations are not an indication of understanding the maths used. Correct algebraic
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© International Baccalaureate Organization 2023
May 2023 subject report
Mathematics: applications and interpretation SL TZ2
manipulation without explanations given for the need of doing so, is not an indication of good
understanding.
Recommendations and guidance for the teaching of future students
Students would benefit from seeing and grading sample explorations available in the Teacher Support
Material and attempting mini-IAs throughout the course.
Additionally, teachers are also encouraged to guide students on how to comment on the validity of the
mathematical processes and data as well as how to reflect meaningfully on the results through use of the
example explorations available in the Teacher Support Material.
Criterion B is a criterion where all students could possibly score well. To make this possible, students
should be given opportunities during the course to practise notation using the appropriate software.
There are many free graphing programmes and teachers should provide the opportunity for students to
learn how to use these effectively. Students could be given a list of appropriate mathematical terms and
their definitions to be used as reference.
Further comments
As stated in all previous reports, it is very important that teachers mark and annotate the submitted
explorations. Errors should be indicated, and comments should be made on the student work justifying
where criterion levels were obtained. Teacher comments justifying the levels awarded should be more
than simply rewording of the criteria.
The exploration should be double spaced and should contain a page count which excludes the title page
and the appendices. Pages should be numbered.
Teachers should double check that the exploration uploaded to IBIS is correct. There were too many
explorations seen where all the mathematical expressions were missing, and the diagrams shifted. This is
to the detriment of the student. In some of the explorations the teacher annotations had also shifted
obscuring the student work. Examiners interact with the scanned work as an image and hence cannot
move teacher comments if they obscure student work.
the marking software for the moderators. This makes it very difficult to match the comments with the
number that appears on the exploration as the moderator must scroll back up every time to see what the
teacher wrote.
Providing a template for statistical explorations were more prevalent this session and should be avoided
as this usually prevents students achieving higher levels in criteria C, D and E.
If there is more than one teacher marking explorations in the school, then internal moderation must take
place to ensure consistency in the application of the criteria.
Lastly, it is very important that no personal details such as the name of the teacher/student or the name
of the school is included in any of the documentation.
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© International Baccalaureate Organization 2023