The use of diagnostic software in teaching a mathematics module for computer science students

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The use of diagnostic software in teaching a
mathematics module for computer science students

• Neil Gordon
• Department of Computer Science
• University of Hull, Hull
• HU6 7RX England
• n.a.gordon_at_hull.ac.uk
• http//www.hull.ac.uk/php/cssnag/

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Brief plan to the talk

• Identify the need for mathematics and formalism
  in computing
• Establish the basis of the problem in
  pre-university mathematics that is creating an
  issue for computing departments
• Consider one approach to dealing with this based
  on using diagnostic formative assessment to drive
  student learning

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Introduction

• The close relationship between mathematics and
  computing as disciplines is well known.
• Recent changes in English mathematics teaching
  and assessment, combined with a decline in the
  basic mathematical skills of students arriving at
  universities is leading to growing difficulties
  for computer science.
• Whilst focussing on the situation in English
  H.E., much of this is relevant in a wider context
  e.g. the problem has been identified
  internationally over several years, for example
  in the U.S.

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The role of mathematics in computing

• Mathematics is naturally occurring in science
  subjects, especially computing. The role of
  mathematics as a key tool has been noted over the
  years.
• Regarding computing, mathematics is identified in
  the subject benchmark, and is specified by many
  professional accrediting bodies (e.g. the BCS).
• Successful teaching of mathematics for computing
  requires that students are able to cope with the
  language and methods of various mathematics
  topics.
• Hence the joint ICS/MSOR meeting on Mathematics
  for Computing

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Admissions requirements

• Historically many computing departments required
  A-level Mathematics for entry
• This is no longer the case, with a wide variety
  of admissions requirements (see ICS network
  survey results)
• However, the perception of mathematics as an
  indicator of computing ability persists

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Weak mathematics weak computing?

• A particular mathematics topic which seems to
  have suffered in recent times is basic algebraic
  manipulation - which is crucial to computing.
• Evidence of the importance of these skills in
  computing is provided by the identification of
  Mathematics and Formalism Education as one of the
  grand challenges facing computing at the 2004
  U.K. Grand Challenges in computing meeting.

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The Gap between university expectations and
students maths skills

• staff in Higher Education departments that rely
  on students having mathematical skills have been
  identifying problems with students grasp and
  fluency in basic maths
• Topics such as algebra, trigonometry and basic
  mathematical manipulation have seemed to be more
  and more problematic for students entering Higher
  Education.
• At Hull we have used diagnostic testing to assess
  these skills for incoming students. Analysis of
  this over the last four years, has revealed a
  measurable decline

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Declining maths skills?
Average overall mark for new students on our
mathematics diagnostic test.
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Interpreting these changes

• These results do not mean students are less able
  mathematically
• However, they identify a growing mismatch between
  University expectations and requirements and
  students own knowledge and skills
• N.B. Mathematics allows objective measurements of
  this discrepancy measured here using a
  diagnostic computer package Diagnosys

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Mathematics Difficulties

• These are caused by a number of distinct isuses
• Widening participation
• larger cohort sizes
• pre-university mathematics experience -
  particularly for students who have only done GCSE
  mathematics - can lead to mathematical illiteracy
  a lack of familiarity with the notations and key
  methods of mathematics
• Mathphobia i.e. the fear of mathematics. can
  lead to mathematical illiteracy, or possibly be a
  consequence of it
• Dyscalculia - is used for those who have a
  difficulty with mathematics due to a learning
  disability similar to dyslexia also known as
  developmental arithmetic disorder, and affects up
  to 6 of children. Dyscalculia is also a
  possible cause of mathphobia

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Diagnostic testing in practice

• Overall average grades in this test are now down
  to 37
• students who have no advanced mathematics grades
  getting an average of only 19
• since 2000, results for students with A-level
  mathematics have improved slightly, but they take
  longer to complete the test
• For all other groups of students, attainment in
  the test is declining.
• This may be explained by the fact that the
  choices provided via AS-levels means that
  students who are weaker at mathematics have the
  option to drop it before A-level.
• The effectiveness of the diagnostic test can be
  seen by the general correlation between
  diagnostic score and incoming mathematics grades

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Scatter plot of incoming mathematics grades
against diagnostic grade (scaled between 0 to 20
where 20 is A in GCSE mathematics, and A in
A-level mathematics, 10 would be GCSE and AS
maths)
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Diagnostic test with Diagnosys

• A computer based mathematics diagnostic
  environment.
• Uses an adaptive skills net to test skills
  efficiently and quickly
• Provides group profiles, with overall grades etc.
  and number of students able to demonstrate
  various mathematical skills
• Supported by the Study Advices Services
• See http//www.staff.ncl.ac.uk/john.appleby/diagpa
  ge/diagindx.htm

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Use of diagnostic results

• Diagnostic tests such as this really require post
  test support.
• Evidence indicates that immediate remedial
  support can help, but upon removal grades
  generally again slip
• Diagnosys gives profiles of individual students
  (which can be returned to them via tutors with
  supporting advice)
• Also provides profiles on the entire class
  allowing for alteration of lecture material etc.
  and inclusion of items where there were common
  problems.

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Diagnostic tests of Hull computing students
2000-2003
?Note the AS mathematics in 2000 was very
different to the new AS levels introduced in 2001.
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Skills against percentage able to do them at
induction
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A framework for supporting mathematics learning

• Pre-module diagnostic test AND appropriate action
  and support within lectures
• Lectures (informed by diagnostic results) and
  closely linked with the main subject
• Workshops with formative assignments to develop
  skills
• Online support materials
• Lecture notes
• Interactive worksheets (MathCAD)
• Links to other support sites
• Liaison and work with Study Advice Services
• Worksheets/special support booklet
• Organised
• Summative coursework to assess progress and
  encourage students engagement with material
• Final exam to assess learning outcomes

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The framework in practice

• Applied to a first year (level 4) quantitative
  methods for computing module
• Taken by students with no advanced mathematics
  (so potentially GCSE C)
• supports a range of modules and degrees in
  computing, ranging from databases in IE, to the
  formalizations required in SE.
• Subjects include set theory relations and
  functions logic algebra trigonometry finite
  state machines vectors and matrices.

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Teaching on the module

• Content influenced by results of diagnostic test
• Include relevant applications of notions in
  computing context essential for students who
  are less secure in their maths
• Use workshops to provide practice (like
  programming, you need to learn by application)

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Computer resources

• Diagnosys test (available for students to retake)
• Module website (lecture notes etc)
• Interactive (MathCAD) worksheets to allow
  interaction with live mathematics
• Links to external support sites (e.g. mathtutor)
• Usual dept. support email/forums/study advice
  resources

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Example MathCAD worksheet. Students are
encouraged to explore and experiment with the
mathematics
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Assessment Program

• assessment designed to differentiate between
  abilities - to provide a challenge for those
  already skilled and able in mathematics, as well
  as encouraging and be accessible to those who
  have less developed mathematical skills.
• Weekly formative assessment covers the main
  concepts met in lectures, with supporting
  workshops
• model solutions released week after, allowing
  more chance for feedback in that weeks workshops.
• module has been run using 3 summative
  assignments, designed to encourage students to
  engage with the material. 2 as exercise sheets, 1
  as a class test
• Final end of module exam (60 of module)

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Evidence of success?

• There are a variety of indicators of success
• Negative indicators include poor attendance at
  workshops take up of the formative worksheets is
  low attendance at the extra-departmental
  workshops became so low that these were stopped
  algebraic and numerical skills in concurrent
  modules showed concerning gaps in students grasp
  of basic mathematical techniques.
• Positive indicators include student feedback that
  many enjoy the material. In fact, several asked
  about studying more mathematics outside of the
  module.
• Exam results mirror other modules in department
  and shows good student achievement with
  acceptable pass rates and average for the module
  being around 50 module mark, with 80 of the
  class passing.

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-Scatter plot of diagnostic score at entry
against module mark -shows that there is little
correlation between incoming maths skills and
final marks in this module -indicates that the
module is successful in providing students with
sufficient maths to overcome any initial barriers
to success in their computer science.
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Conclusions the way ahead?

• We have considered a framework for supporting
  students in studying mathematics for computing.
• Along with identification of the wider context
  and problems that affect the learning of the
  subject, we have some possible approaches to help
  students
• Further development of a suitable framework
  include
• embedding the diagnostic test as a formal part of
  assessment
• Encourage students to attend workshops by
  embedding coursework into weekly worksheets
• Formal evaluation of the approach
• Since mathematics skills are crucial to a
  complete computing education, teaching these to
  modern computing students is an important but
  continually changing task.