Brief History of Math-Bridge and its Usage

Brief History of Math-Bridge and its
Usage
George Goguadze
Leuphana Universität Lüneburg

Math-Bridge: short summary of features
▪ Tools for students:
▪ Static Courses
▪ Adaptive Course Generation
▪ Micro Course Generation
▪ Intelligent search
▪ Interactive exercises
▪ Student progress indicators
▪ What is the target audience?
▪ How do students work with the
system?
▪ Which features are useful?
▪ Which pedagogical approaches
are suitable?

Math-Bridge is a family of technologies made by a community
of researchers
ActiveMath-‐
EU
Active
Math
Le-‐
ActiveM
ath
Math-‐
Bridge
Math-‐Bridge+

MathCoach
ATuF
2
2000
2004
Matheführer
schein
ATuF
ALOE
Mathe-‐
Brücke
Math-‐Bridge

final
2007
2009 2011
2013
?
eChalk

First Version of ActiveMath
• Course
Generation

• User
Model

• Progress
indicators

• Multiple
Choice

Questions

• CAS
exercises

First Version of ActiveMath
• Next
Best

Suggestion
Engine

• „Poor
man‘s“
eye-‐

tracker
„DFKeye“

Second Version of ActiveMath
• Dashboard-‐style
main
screen

• „better“
design

• Open
User
Model

Math-Bridge is a family of technologies made by a community
of researchers
ActiveMath-‐
EU
Active
Math
Le-‐
ActiveM
ath
Math-‐
Bridge
Math-‐Bridge+

MathCoach
AtUF
2
2000
2004
Matheführer
schein
AtUF
ALOE
Mathe-‐
Brücke
Math-‐Bridge

final
2007
2009 2011
2013
?
eChalk

Matheführerschein (Driving License for Math)
▪ Matheführerschein Online helps preparing learners for college/
university
▪ It was initiated by FH Dortmund
▪ Content: wide range of school mathematics needed for University
▪ Fractions, Equations, Term Manipulation, Functions,
Differentiation and Integration
▪ The pedagogical approach is constructivist, starting from complex
real world problems
▪ ActiveMath interface was modified
▪ Specific strategy for interactive exercises was developed
▪ The ActiveMath possessed a library of terms with a novel structure
(flavours and links to exercises)
!
!

Matheführerschein
• The
system
received
positive

reviews
for
its
intuitive
design
and

didactic
approach

• Matheführerschein
is
available

online
for
wide
public

• Freshly
enlisted
students
from
FH

Dortmund
were
recommended
to

use
the
system

• Hundreds
of
Students
tested
their

knowledge
using
the
system

• No
records
of
learning
effect

!

eChalk: Algebraic Geometry on a Smartboard
▪ This project connects three systems:
▪ ActiveMath
▪ Computer Algebra System
Singular
▪ Smartboard technology eChalk
▪ A Course of Algebraic Geometry
given by Prof. Schreyer
▪ Contents of a Course Book
encoded in ActiveMath
▪ Handwriting recognition and
Computer Algebra work together
to help the lecturer manipulate
interactive visualizations
▪ The Course was held at the
Mathematics Faculty at Saarland
University

Le-ActiveMath: The next generation

Features of Le-ActiveMath
• Tutorial
Dialogue
in
exercises

• Better
design

• Concept
Mapping
Tool

• Elaborate
pedagogical
scenarios

Features of Le-ActiveMath
• „Scrutable“
User
Model

Example Usage of Le-ActiveMath
Practical
Calculus
Course

!
238 Practical Calculus students, Edinburgh University, UK.
Mean Age = 19
!
• 11
week
course,
1
tutorial
every
fortnight.

• Traditionally,
tutorials
and
homework
paper-‐
based.

• Le
Active
Math
used
instead.

• Content:
University
first
year
Calculus

• Pre-‐recorded
books
authored
for
course
and
each

homework/tutorial.

à LeAM
could
be
used
in
3
homework/
tutorials.
Source:

Tim
Smith

Tutorial Structure
▪ Before Tutorial
▪ Lecturer sets homework on LeActiveMath.
▪ Students complete exercises at home/computer lab.
▪ Answers automatically logged.
▪ Links in LeActiveMath between homework content and other
content assist students.
▪ In preparation
▪ Admin Pages developed to allow tutors to view student progress.
▪ Reporting Tool produces reports on a group of user’s attempts at
exercises.
▪ In Tutorial
▪ Students completed exercises and browsed the content if needed
Source:

Tim
Smith

LeActiveMath Usage
▪ Expected regular usage
▪ Peak of usage prior to tutorials
▪ Increasing mean usage prior to exam.
▪ Observed very low usage.
▪ Most users were those recruited for in-depth tasks.
▪ Usage was mostly on exercises and searching for content.
▪ Advanced components rarely used.
!
▪ There were also some technical problems, so the usage
statistics is not reliable
Source:

Tim
Smith

▪Users found navigation of the content easy.
▪Users liked the book metaphor, search tool, and
hyperlinks.
▪But
▪ The content was often confusing, too scattered with jargon, and
the difficulty level incorrect.
▪ They found the search tool too complicated e.g. they had to
select too many options to find the content in a book.
In-depth Evaluation: Summary
Source:

Tim
Smith

In-depth Evaluation: Summary
▪Formula Editor
▪ Is seen as a useful tool
▪ But it is unintuitive, too particular in its syntax, and frustrates
users.
▪Hints and feedback
▪ Learners find them useful
▪ But could have more levels of hints and always bottom out at
solution.
▪Exercise Types
▪ Learners see the benefit of most types of exercises
▪ But prefer MCQ, SCQ, and computations without the input editor.
Source:

Tim
Smith

ActiveMath-EU: Dissemination and Exploitation of LeActiveMath

ActiveMath-EU: Using LeActiveMath in classroom
▪ ActiveMath usage for multilingual pre-service teachers
▪ In Charles University in Prague for pre-service teacher students learning math in
Czech and English in parallel
▪ In Eötvös Lorand University in Budapest for pre-service teacher students learning
math in Hungarian and German
▪ Other Sample Usage Scenarios
▪ Blended Learning in a classroom moderated by a teacher in Eötvös Lorand
University Budapest
▪ Solving interactive Exercises in a secondary school in Germany
▪ Blended Learning and learning assignments with particular learning paths for pre-
service teachers in Université Pierre Marie Curie, Paris 6
▪ Blended-Learning with homework assignments at St. Michael College in the
Netherlands (own content)

ALOE Project
• ALOE
project
investigated
the
Effects
of
erroneous
examples
in
the

domain
of
decimals

• Several
school
experiments
were
conducted
in
Germany
and
U.S.

• 6th
grade,
7th
grade,
and
8th
grade

• Interactive
exercises
were
solved
in
a
classroom
in

teacher-‐
assisted
exercise
sessions

• Pupils
worked
with
fixed
sequences
of
learning
objects

• General
comments
on
the
usage
of
the
system

• After
just
1
hour
of
familiarization,
pupils
are
able
to
cope
with

the
system
navigation
and
formula
input

• Intuitive
user
interfaces
are
important
for
school
context

• Good
observations:

• Students
find
and
describe
errors,
but
cannot
correct

➢declarative
vs.
practical
knowledge

• Students
solve
similar
exercises,
but
cannot
correct
errors

➢Memorized
solution
practice,
but
lack
of
deeper

knowledge

!
!

School Fraction Course and ATuF Project
• A
Fraction
Course
for
School
was
authored
in
ActiveMath
by

a
teacher
(Mr.
Kessler)

• He
used
this
course
for
teaching
fractions
in
a
secondary

school
in
Saarbrücken
for
2
Semesters

• The
Course
was
further
reworked
within
the
DFG
Project

ATuF
(Adaptive
Tutorial
Feedback)

• ATuF
investigates
various
feedback
strategies
for
interactive

exercises,
based
on
the
feedback
framework
of
Prof.
S.

Narciss

• A
structured
user
interface
for
solving
interactive
exercises
in

the
domain
of
fractions
was
developed

• Several
feedback
strategies
have
been
tested
with
students

• Lab
experiments
with
about
200
students
were
conducted

!
!
!

Math-Bridge: Intelligent Remedial Mathematics
• The
goal
of
the
project
was
to
create
a
European
portal
for

mathematical
bridging
courses

• The
final
product
should
be
disseminated
to
the
educational

institutions
and
used
for
teaching

• Project
partners
have
used
the
system
for
teaching
in
their

institutions,
there
is
a
community
of
associate
partners

• Leading
Universities
in
Europe
and
Industrial
Partners
and
Sub-‐
contractors
have
contributed
to
the
project

Some Users of Math-Bridge
▪ Math-Bridge was used in bridging courses at Eötvös Lorand University
for pre-service teachers
▪ HTW Saarland uses Math-Bridge in combination with Math-Coach
System
▪ Universities of Kassel and Paderborn used Math-Bridge for
Mathematics bridging courses for technical faculties
▪ University of Brandenburg used Math-Bridge for their Mathematics
bridging course for Computer Scientists
▪ Math-Bridge is currently used at Leuphana University of Lüneburg in a
Mathematics bridging course for economists.

Mathematics Bridging Course at Leuphana University
Bridging-‐
course

(7
Weeks)
Pretest
Posttest
Blended-‐Learning

Bridging
Course

(7
Weeks)
Blended-‐Learning

Bridging
Course

(4
Weeks)
Lecture–
Mathematics
for

Economics
Exam
Repeated

Exam
The
first
semester
at
Leuphana
University
(so-‐called
Leuphana
Semester)
is

divided
into
two
halves:

!
• The
first
half
is
devoted
to
introductory
courses
and
bridging
courses

• In
the
second
half
some
major
Mathematics
lectures
build
upon
the

introductory
courses

Bridging-‐
course

(7
Weeks)
Pretest
Posttest
Blended-‐Learning

Bridging
Course

(7
Weeks)
Blended-‐Learning

Bridging
Course

(4
Weeks)
Lecture–
Mathematics
for

Economics
Exam
Repeated

Exam
• Pretest
determines
the
knowledge
gaps

• Bridging
course
in
the
first
7
weeks
is
using
math-‐bridge
and
other
technologies

• Math-‐Bridge
is
used
for
information
and
training
at
home

• Other
teacher
tools
are
used
during
the
classes

• Lecture
materials
are
linked
to
Math-‐Bridge

• Posttest
shows
the
improvement
for
those
who
attended
the
first
bridging

course

Bridging-‐
course

(7
Weeks)
Pretest
Posttest
Blended-‐Learning

Bridging
Course

(7
Weeks)
Blended-‐Learning

Bridging
Course

(4
Weeks)
Lecture–
Mathematics
for

Economics
Exam
Repeated

Exam
• A
blended
learning
bridging
course
using
Math-‐Bridge
is
given
in
the
next
7

weeks

• The
main
Mathematics
lecture
is
running
in
parallel
to
the
second
bridging

course,
offering
the
students
with
difficulties
to
join
right
away
and
train
with

Math-‐Bridge
system.

• Another
intensive
blended-‐learning
course
using
Math-‐Bridge
is
offered

between
two
exams
(duration
4
weeks)

How to teach it? Old and new Technologies for Mathematics
Lecture
24.09.14 40
• Conflict:
Blackboard
vs.
Computer
&
Projector

• Blackboard:

• Chalk
supports
arbitrary
formula
input
and
visualizations

• Full
freedom
for
improvisation
with
examples

• Computer:

• Power
Point
Slides:
complex
diagrams
and
animations

• GeoGebra
Animations:
examples
to
touch

• Intelligent
Computer
Algebra
Systems

• Commonly
used
solutions:

• Use
Smartboards
to
combine
blackboard
and
computer

• Use
tablet
computers
connected
to
a
projector

Our solution: E-Learning / E-Teaching Technologies
!
‣ Structured interfaces for teacher to interact with presented content
‣ Termania – tool for visualizing term manipulation
‣ GeoGebra
‣ Intelligent Learning Environment Math-Bridge
‣ VEMINT Portal Contents

First Bridging Course: Structure & Learning Materials 
24.09.14 42
!
• Course:
Bridging
Course
Mathematics
for
Economics

• Number
of
students:
max
250

• Lecture

• Two
times
a
week
two
hours
each
time

• One
book
chapter
per
week
(in
total
6
Chapters)

• Power
Point
Slides

• Animations
in
the
slides

• External
Animations

• Learning
materials

• Book
„Mathematics
for
Economics“
Chapters
1-‐6

• Slides
including
video
recorded
animations
from
the

lecture

• Additional
materials
in
Math-‐Bridge

• Animations
in
Youtube
und
GeoGebra
portals
as
extra

channels

Self-learning & Social Learning
▪ Micro-prelearning (Math-Bridge):
▪ Animated worked solutions (Math-Bridge, youtube)
▪ Training exercises with feedback (similar to homework exercises)
▪ Self-learning (Math-Bridge)
▪ Working with additional materials:
▪ Math-Bridge books, interactive exercises, (micro) course generation
▪ Social Browsing:
▪ Youtube channel of Math-Bridge
▪ One can browse related videos brought by youtube keyword matching
▪ Students can add own videos
▪ Browse GeoGebra Animation portal
▪ Animations from the lecture are uploaded and linked to
▪ Similar animations from GeoGebra portal can be browsed, they are
automatically linked by common keywords
24.09.14 43

Math-Bridge Contents
▪ Each content book corresponds to a chapter of the course book
▪ The structure of each chapter is fixed and the students are suggested to follow particular
learning paths, depending on their goals

45
Interactive
Power
Point
Slides
=+b +a
+b
a+b=b+a a-b=a+(-b)
+a -‐b -‐b +a
a-(-b)+2a+b-a= a+b+2a+b-a= bereinigen
+a
+a +b =
+2a +b -a
+2a +2b

46
Pascal´s
Triangle
5
2
!
"
#
$
%
& =
5!
2!3!
=
4⋅5
2
=10
5
2
!
"
#
$
%
& =
4
1
!
"
#
$
%
&+
5
2
!
"
#
$
%
& = 4 + 6 =10

Term-manipulation- and Animation tool „Termania“

GeoGebra- Visualization
24.09.14 49

Geogebra Portal: Bridging Course Collection
24.09.14 50
Visualizations
and
animations

of
various
concepts

Youtube channel: Math-Bridge Leuphana
24.09.14 51
Videos
of
Term-‐Manipulation

Blended-Learning Bridging Course (using Math-Bridge)
running in parallel to the main Mathematics Lecture
!
▪ Number of Students: max 60
▪ Course is given in a computer equipped seminar room
▪ Topics:
▪ Static and Personalized Courses in Math-Bridge corresponding
the the contents of Chapters 1-6 of the reference book
▪ Visualizations and Examples
▪ Special examples for economists
▪ Animations (GeoGebra)
▪ Exercises:
▪ Demonstration of worked solutions
▪ interactive exercises in Math-Bridge
24.09.14 52

Blended-Learning Scenario: „Teach & Train“
!
‣ Introduction to a lecture topic
‣ Definitions, interactive examples using Termania and GeoGebra
!
!
‣ Solving problems in Math-Bridge
‣ Interactive exercises with feedback and hints
‣ Self-assessment exercises solved on the paper and compared to the
master solution in the system
‣ Multimedial exercises correct/wrong feedback

Students about using Math-Bridge
▪ Course Materials are easy to browse, intuitive and well structured
▪ However
▪ Most of the students did not notice that the search function exists
▪ Not all chapters of static books were structured in the same way, which
introduced some confusion
▪ Most of the students did not use the micro course generation feature, even after
explicitly introducing them to the feature
▪ Solving interactive exercises was too much effort for some students
▪ Formula editor is complex and slow means of entering formulas
▪ Many students prefer to write the solution of the paper and submit the final
result into the system
▪ Multiple choice questions, graphical puzzles and one step exercises with simple
input were more popular

Further Steps
!
‣ Automatic Generation of Termania Examples
‣ How?
‣ Use integrated domain reasoners to generate the annotated
solutions
‣ Why do we need this?
‣ The teacher can generate on the fly a worked solution of an
improvised example and show it right away
‣ Automatic Integration of the lecture slides in the system
‣ Generation of annotated e-Lectures
‣ Automatic annotation of parts of videos with links to corresponding
concepts
‣ Evaluation of the learning effect of the „Teach & Train“
strategy and comparison to the classical lecture

24.09.14 57
Classroom of the future (1969, Shōnen Sunday magazine)
E-‐Lecture
Working
with

Math-‐Bridge
Flag

Feedback

24.09.14 58
Classroom of the future: the future is NOW!

Thanks!
The
Mathematical
Bridge,
Cambridge,
England

Brief History of Math-Bridge and its Usage

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