Mathematics in English at HHX - Focus Points and Possible Solutions
Ingrid
Casse
Ingvar
Hansson
Benjamin D.
Heredia
Matematik på
Engelsk
Mathematics in English
at
HHX…
Focus
Points
E
n
g
l
i
s
h
a
s
l
a
n
g
u
a
g
e
V
ocabulary
Variation
T
echnical
T
eaching
Topics
No
corresponding
English
textbook
No direct
correlation
between
English
textbooks
and
Danish
textbooks
Mathematics in English
at
HHX…
Focus
Points
E
n
g
l
i
s
h
a
s
l
a
n
g
u
a
g
e
Student
background
Variation
in
student’s
language
skills
New
vocabulary
Inter-
disciplinary
work
Business
Economics
Marketing
International
economics
Evt. other
subjects.
Mathematics in English
at
HHX…
Focus
Points
E
n
g
l
i
s
h
a
s
l
a
n
g
u
a
g
e
UK vs.
US
Mathematic
symbols,
i.e.
uses of
letters
Europe
vs.
American
tradition
T
erminology
T
ime
Preparation (e.g.
videos)
Finding
of
material
Formulation
of
examples
and
assignments
Finding
right level
and
relevancy
Mathematics in English
at
HHX…
T
eaching
Topics
No
corresponding
English
textbook
No direct
correlation
between
English
textbooks
and
Danish
textbooks
Possible
Solutions
Develop own
material
Select
chapters
(or
parts
of)
from
various
sources
Websites,
videos
e.g.
zweigmedia.com
,
sosmath.com
,
mathsisfun.com
,
youtube.com
,
khanacademy.org
Mathematics in English
at
HHX…
Possible
Solutions
e.g.
zweigmedia.com
,
sosmath.com
,
mathsisfun.com
Mathematics in English
at
HHX…
Possible
Solutions
e.g.
youtube.com
,
khanacademy.org
,
nancypi.com
Mathematics in English
at
HHX…
Possible
Solutions
e.g.
youtube.com
,
khanacademy.org
,
nancypi.com
Mathematics in English
at
HHX…
Possible
Solutions
e.g.
books and
chapters (parts
of)…
Mathematics in English
at
HHX…
Possible
Solutions
e.g.
books and
chapters (parts
of)…
Mathematics in English
at
HHX…
Possible
Solutions
e.g.
books and
chapters (parts
of)…
Mathematics in English
at
HHX…
Possible Solutions: Example
Math-C
Start: youtube
material
Mathematics in English
at
HHX…
Possible
Solutions
Own material (quadratic equations
and second degree
polynomials)
A
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t
i
v
i
t
y
:
I
n
G
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e
p
o
s
i
t
i
o
n
o
f
t
h
e
v
e
r
t
e
x
.
Analyse
coefficients
by moving
the
sliders
Second
degree
(quadratic)
polynomials
2
f
(
x
)
=
ax
+
bx
+
c
a≠0
y
x
M
i
n
d
t
h
e
t
e
r
m
i
n
o
l
o
g
y
:
C
o
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c
a
v
e
d
o
w
n
w
a
r
d
s
!
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p
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:
P
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a
Second
degree
(quadratic)
T
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:
V
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y-intercept
ze
r
o
ze
r
o
f
(
x
)
=
ax
2
+
bx
+
c
a≠0
polynomials
M
i
n
d
t
h
e
t
e
r
m
i
n
o
l
o
g
y
:
C
o
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c
a
v
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u
p
w
a
r
d
s
!
y
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-
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(
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q
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o
n
)
x
Mathematics in English
at
HHX…
Possible
Solutions
Own material (quadratic equations
and second degree
polynomials)
M
i
n
d
t
h
e
l
a
n
g
u
a
g
e
:
-
We
know
that
solve
the
equation:
at any point on
the
x
-axis,
therefore
we
must
,
-
We
use
the
discriminant
to calculate the
zeros
-
The formula
of
the
discriminant
is:
Rules of
the
value of
d
:
When
d>0,
there
are
two
real
zeros
When
d=0,
there
is one real zero
When
d>0,
there
are no real
zeros
Mathematics in English
at
HHX…
Possible
Solutions
e.g.
own
material (quadratic equations
and second degree
polynomials)
x
=
b
±
p
b
2
4
ac
2
a
Z
e
r
o
s
(
r
o
o
t
s
)
:
T
h
e
f
r
a
c
t
i
o
n
i
n
v
o
l
v
e
s
a
n
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r
a
t
o
r
w
i
t
h
a
s
q
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e
r
o
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t
,
w
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a
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.
V
e
r
t
e
x
:
T
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t
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x
i
s
a
n
o
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d
p
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c
o
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t
a
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n
i
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g
a
f
i
r
s
t
-
c
o
o
r
d
i
n
a
t
e
(
x
)
a
n
d
a
s
e
c
o
n
d
-
c
o
o
r
d
i
n
a
t
e
(
y
)
.
a
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e
p
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o
m
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f
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a
n
t
Mathematics in English
at
HHX…
Possible
Solutions
e.g.
own
material (quadratic equations
and second degree
polynomials)
A
p
p
l
i
c
a
t
i
o
n
s
The
demand
for the products
of a company is given by
the
linear
price/demand
function:
is
the
price
;
x
is
the units
sold,
T
h
e
r
e
v
e
n
u
e
o
f
t
h
e
c
o
m
p
a
n
y
i
s
g
i
v
e
n
b
y
:
T
h
e
r
e
f
o
r
e
t
h
e
r
e
v
e
n
u
e
f
u
n
c
t
i
o
n
b
e
c
o
m
e
s
:
which is a
quadratic
function.
a<0 so
the function
is concave downwards,
a
and
b
have
opposite
algebraic
signs so
the vertex sits to the
right of
the
y-axis,
c
is 0 so
the y-intercepts
is at 0
TASK:
Find
maximum revenue and account
for the revenue’s
domain.
Mathematics in English
at
HHX…
Possible Solutions: Example
Math-B
e.g.
books and
chapters (parts
of)…
Linear
programming
(simplex
method).
g.
L
i
n
e
a
r
p
r
o
g
r
a
m
m
i
n
g
(
g
e
o
m
e
t
r
i
c
m
e
t
h
o
d
)
:
-
C
o
m
m
o
n
l
a
n
g
u
a
g
e
-
Objective functions
-
Systems
of
inequalities
(constraints)
-
Nonnegative constraints
-
Feasible
regions
(bounded,
unbounded)
-
Feasible solutions
-
Corner
points (solutions
of
linear
systems
of
inequalities)
-
Optimum
solution
-
Isolines
Mathematics in English
at
HHX…
Possible Solutions: Example
Math-B
g.
L
i
n
e
a
r
p
r
o
g
r
a
m
m
i
n
g
(
g
e
o
m
e
t
r
i
c
m
e
t
h
o
d
)
:
-
S
u
g
g
e
s
t
i
n
g
a
s
t
r
a
t
e
g
y
:
-
Step
1. Decision
variables,
-
Step
2. Summarise relevant
information
in
tabular form
and
relate to
decision
variables,
-
Step
3.
Determine the objective
(maximise or minimise) and
establish the objective
function,
-
Step
4.
Write
problem
constraints
and
find
corner
points,
-
Step
5. Draw
feasible
region
and
establish
isolines,
-
Step
6.
Determine optimum
solution
and
conclude.
Mathematics in English
at
HHX…
Possible Solutions: Example
Math-B
Mathematics in English
at
HHX…
Possible Solutions: Example
Math-A
E
.
g
.
Q
u
a
d
r
a
t
i
c
p
r
o
g
r
a
m
m
i
n
g
(
g
e
o
m
e
t
r
i
c
m
e
t
h
o
d
)
:
-
C
o
m
m
o
n
l
a
n
g
u
a
g
e
(
k
n
o
w
n
f
r
o
m
f
i
r
s
t
y
e
a
r
,
i
.
e
.
l
i
n
e
a
r
p
r
o
g
r
a
m
m
i
n
g
)
-
Conic
sections
-
L
e
v
e
l
c
u
r
v
e
s
,
c
o
n
t
o
u
r
m
a
p
s
-
Constraints
and
nonnegative
constraints
M a t L a
b
Top:
MatLab contour maps and
horizontal
traces of raising level curves for
function:
Bottom: same
function
plotted in
GeoGebra
G
e
o
G
e
b
r
a
Mathematics in English
at
HHX…
Possible Solutions: Example
Math-A
E
.
g
.
Q
u
a
d
r
a
t
i
c
p
r
o
g
r
a
m
m
i
n
g
(
g
e
o
m
e
t
r
i
c
m
e
t
h
o
d
)
:
-
C
o
m
m
o
n
l
a
n
g
u
a
g
e
(
k
n
o
w
n
f
r
o
m
f
i
r
s
t
y
e
a
r
,
i
.
e
.
l
i
n
e
a
r
p
r
o
g
r
a
m
m
i
n
g
)
-
Conic
sections
(e.g.
parabola,
ellipse)
-
Level curves,
contour
maps
-
Constraints
and
nonnegative constraints
G
e
o
G
e
b
r
a
2D and 3D
picture
of
function
A
s
s
o
c
i
a
t
e
d
t
o
t
h
e
c
o
n
s
t
r
a
i
n
t
:
Mathematics in English
at
HHX…
Possible
Solutions
e.g.
development of own
materials…
Mathematics in English
at
HHX…
Possible
Solutions
e.g.
use of previous
examination sets for writing
skills.
Mathematics in English
at
HHX…
Possible
Solutions
e.g. student
activity…
Mathematics
Project
(open
access)
Mathematics in English
at
HHX…
Possible
Solutions
e.g. student assignments…
Videos
Mathematics in English
at
HHX…
Possible
Solutions
e.g. student
activity…
Notes
Mathematics in English
at
HHX…
Possible
Solutions
e.g. student
activity…
Assignments
-
Math
C
Factorise following
functions
a)
f
(
x
)
x
2
3
x
2
b)
f
(
x
)
2
x
2
5
x
2
c)
f
(
x
)
x
2
7
x
6
d)
f
(
x
)
3
x
2
3
x
6
e)
f
(
x
)
3
x
2
6
x
9
f)
f
(
x
)
8
x
2
6
x
1
Determine the
discriminant
d
and
the
zeros
for
all
the
functions
Mathematics in English
at
HHX…
Possible
Solutions
e.g. student
activity…
Assignments
-
Math
B
Mathematics in English
at
HHX…
Possible
Solutions
g.
student
activity…
Assignments
-
Math
A
F
o
r
t
h
e
p
a
r
a
b
o
l
a
,
t
h
e
s
e
t
o
f
p
o
i
n
t
s
i
n
t
h
e
c
u
r
v
e
f
r
o
m
a
f
i
x
e
d
p
o
i
n
t
l
o
c
a
t
e
d
a
l
o
n
g
t
h
e
a
x
i
s
o
f
s
y
m
m
e
t
r
y
(
v
i
z
.
f
o
c
u
s
)
,
i
s
e
q
u
i
d
i
s
t
a
n
t
t
o
a
f
i
x
e
d
l
i
n
e
(
v
i
z
.
d
i
r
e
c
t
r
i
x
)
.
A
s
s
u
m
e
t
h
a
t
a
n
e
q
u
a
t
i
o
n
o
f
t
h
e
p
a
r
a
b
o
l
a
w
i
t
h
f
o
c
u
s
(
0
,
c
)
a
n
d
d
i
r
e
c
t
r
i
x
i
s
:
S
k
e
t
c
h
a
n
d
a
n
a
l
y
s
e
f
o
l
l
o
w
i
n
g
p
a
r
a
b
o
l
a
s
:
a)
when
b)
when
c)
when
d)
when
e)
find the focus and
directrix
for
f)
find the equation of the parabola with
vertex
(0, 0) and focus (5,
0).
g)
let the
standard
equation of a parabola
with
vertex
and
vertical
axis of
symmetry
be
, find the focus and the
directrix.
h)
find the
vertex,
the focus and the
directrix
for
(HINT:
rewrite
in
standard
form
).
i) Designers are
testing
satellite dish with parabolic cross
sections. The
dish design is 150 cm wide
at
the opening and the focus is placed 40 cm from the
vertex.
Your
tasks are: 1) draw the dish by
positioning a coordinate
system
with origin at the
vertex
and x-axis along the
parabola’s
axis of
symmetry; 2) find an equation for the parabola; 3) find the depth of the satellite dish at the
vertex.
Mathematics in English
at
HHX…
Possible
Solutions
e.g.
books and
chapters (parts
of)…
References for Mathematics
B:
References for Mathematics
A:
Websites:
zweigmedia
.
co
m
,
sosmath.com
,
mathsisfun.com
-
Haese et al. (2012)
Mathematics for the International Student
(3rd.
Edition).
Haese
Mathematics
Ed.
-
Wazir
et al. (2012)
Mathematics
Higher Level (2012
Edition).
Pearson
Edu.
Ltd.
-
Haeussler et al. (2008)
Introductory Mathematical
Analysis
for
Business,
Economics, and
the Life
and Social Sciences
(12th Edition)
Pearson Edu.
Ltd.
-
Barnett
et al. (2015) College
Mathematics for
Business Economics and
Social Sciences
(13th Edition)
Pearson Edu.
Ltd.
Conclusions:
Although focus points
can be
many,
it is
suggested
to:
-
Work
out a progression using a
combination
of
natural
and
mathematical
language
let
the
students
communicate
(brief
presentations)
use
the
mathematics
(e.g.
repetition)
Establish
a
common
language
Involve the
managers
Involve
colleges
(economics
and
languages)
Conclusions:
-
Students
can
misundertand
statements
or cannot connect
former
knowledge
with
same knowledge in
other
language
(i.e.
language
barriers)
-
Finding
relevant
material
is hard and
time-consuming
-
Much in
the internet
(including Khan) is good
for practicing
but lacks
theory
-
Most of
the results
in google are useless so googling
for information
is hard and
time-consuming.
-
Translation
to
Danish
requirements (bekendtgørelsen)
is hard and
time-consuming
The article discusses the focus points and possible solutions for teaching mathematics in English at HHX, highlighting topics like technical English, vocabulary variation, student backgrounds, and terminology differences between UK and US. Possible solutions include developing custom materials, selecting chapters from various sources, and utilizing websites and videos for teaching support.
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Matematik p Engelsk Ingrid Casse Ingvar Hansson Benjamin D. Heredia
Mathematics in English at HHX Focus Points T echnical English as language Vocabulary Variation No corresponding English textbook T eaching Topics No direct correlation between English textbooks and Danish textbooks
Mathematics in English at HHX Focus Points New vocabulary Student background English as language Variation in student s language skills Business Economics Inter- disciplinary work Marketing International economics Evt. other subjects.
Mathematics in English at HHX Focus Points T erminology English as language UK vs. US Mathematic symbols, i.e. uses of letters Europe vs. American tradition Preparation (e.g. videos) Time Finding of material Formulation of examples and assignments Finding right level and relevancy
Mathematics in English at HHX Possible Solutions Develop own material No corresponding English textbook T eaching Topics Select chapters (or parts of) from various sources No direct correlation between English textbooks and Danish textbooks Websites, videos e.g. zweigmedia.com, sosmath.com, mathsisfun.com, youtube.com, khanacademy.org
Mathematics in English at HHX Possible Solutions e.g. zweigmedia.com, sosmath.com, mathsisfun.com
Mathematics in English at HHX Possible Solutions e.g. youtube.com, khanacademy.org, nancypi.com
Mathematics in English at HHX Possible Solutions e.g. youtube.com, khanacademy.org, nancypi.com
Mathematics in English at HHX Possible Solutions e.g. books and chapters (parts of)
Mathematics in English at HHX Possible Solutions e.g. books and chapters (parts of)
Mathematics in English at HHX Possible Solutions e.g. books and chapters (parts of)
Mathematics in English at HHX Possible Solutions: Example Math-C Start: youtube material
Mathematics in English at HHX Possible Solutions Own material (quadratic equations and second degree polynomials) Activity: In GeoGebra, make three sliders, a, b and c and determine from this how the coefficients influence the shape of the parabola and the position of the vertex. Analyse coefficients by moving the sliders
Second degree (quadratic) polynomials 2 f (x) = ax + bx+ c Mind the terminology: Concave downwards! a 0 y The point at which the graph turns over is called vertex This point is also the graph s maximum The axis of symmetry is line the passing through the vertex, dividing the parabola in exactly two parts The graph of a quadratic function: Parabola x
Second degree (quadratic) polynomials f (x) = ax2+ bx+ c a 0 Mind the terminology: Concave upwards! y The graph of a quadratic function: Parabola The axis of symmetry is line the passing through the vertex, dividing the parabola in exact two parts y-intercept x-intercepts are called zeros of the function (or equation) zero zero x The point at which the graph turns over: Vertex This point is also the graph s minimum
Mathematics in English at HHX Possible Solutions Own material (quadratic equations and second degree polynomials) Mind the language: at any point on the x-axis, therefore we must , - We know that solve the equation: - We use the discriminant to calculate the zeros - The formula of the discriminant is: Rules of the value of d: When d>0, there are two real zeros When d=0, there is one real zero When d>0, there are no real zeros
Mathematics in English at HHX Possible Solutions e.g. own material (quadratic equations and second degree polynomials) Zeros (roots): The fraction involves a numerator with a square root, which has always a positive and a negative solution, and a denominator. pb24ac 2a b x = Vertex: The vertex is an ordered pair containing a first-coordinate (x) and a second-coordinate (y). are polynomial coefficients and d is the discriminant
Mathematics in English at HHX Possible Solutions e.g. own material (quadratic equations and second degree polynomials) Applications The demand for the products of a company is given by the linear price/demand function: ; x is the units sold, is the price The revenue of the company is given by: Therefore the revenue function becomes: which is a quadratic function. a<0 so the function is concave downwards, a and b have opposite algebraic signs so the vertex sits to the right of the y-axis, c is 0 so the y-intercepts is at 0 TASK: Find maximum revenue and account for the revenue s domain.
Mathematics in English at HHX Possible Solutions: Example Math-B e.g. books and chapters (parts of) Linear programming (simplex method).
Mathematics in English at HHX Possible Solutions: Example Math-B g.Linear programming (geometric method): - Common language - Objective functions - Systems of inequalities (constraints) - Nonnegative constraints - Feasible regions (bounded, unbounded) - Feasible solutions - Corner points (solutions of linear systems of inequalities) - Optimum solution - Isolines
Mathematics in English at HHX Possible Solutions: Example Math-B g.Linear programming (geometric method): - Suggesting a strategy: - Step 1. Decision variables, - Step 2. Summarise relevant information in tabular form and relate to decision variables, - Step 3. Determine the objective (maximise or minimise) and establish the objective function, - Step 4. Write problem constraints and find corner points, - Step 5. Draw feasible region and establish isolines, - Step 6. Determine optimum solution and conclude.
Mathematics in English at HHX Possible Solutions: Example Math-A E.g. Quadratic programming (geometric method): M a t L a b - Common language (known from first year, i.e. linear programming) - Conic sections - Level curves, contour maps - Constraints and nonnegative constraints Top: MatLab contour maps and horizontal traces of raising level curves forfunction: Bottom: same function plotted in GeoGebra G e o G e b r a
Mathematics in English at HHX Possible Solutions: Example Math-A E.g. Quadratic programming (geometric method): 2D and 3D picture of function Associated to the constraint: - Common language (known from first year, i.e. linear programming) - Conic sections (e.g. parabola, ellipse) - Level curves, contour maps - Constraints and nonnegative constraints G e o G e b r a
Mathematics in English at HHX Possible Solutions e.g. development of own materials
Mathematics in English at HHX Possible Solutions e.g. use of previous examination sets for writing skills.
Mathematics in English at HHX Possible Solutions e.g. student activity Mathematics Project (open access)
Mathematics in English at HHX Possible Solutions e.g. student assignments Videos
Mathematics in English at HHX Possible Solutions e.g. student activity Notes
Mathematics in English at HHX Possible Solutions e.g. student activity Assignments - Math C Factorise following functions a) f(x) = x2 3x+2 b) f(x) = 2x2 5x+2 f(x) = x2+7x 6 d) f(x) = 3x2 3x 6 c) e) f(x) = 3x2+6x +9 f(x) = 8x2 6x+1 f) Determine the discriminant d and the zeros for all the functions
Mathematics in English at HHX Possible Solutions e.g. student activity Assignments - Math B
Mathematics in English at HHX Possible Solutions student activity Assignments - MathA g. For the parabola, the set of points in the curve from a fixed point located along the axis of symmetry (viz. focus), is equidistant to a fixed line (viz. directrix). Assume that an equation of the parabola with focus (0, c) and directrix is: Sketch and analyse following parabolas: a) when b) when c) when d) when e) find the focus and directrix for f) find the equation of the parabola with vertex (0, 0) and focus (5, 0). g)let the standard equation of a parabola with vertex be, find the focus and the directrix. and vertical axis of symmetry h)find the vertex, the focus and the directrix for standard form (HINT: rewrite in ). i) Designers are testing satellite dish with parabolic cross sections. The dish design is 150 cm wide at the opening and the focus is placed 40 cm from the vertex. Your tasks are: 1) draw the dish by positioning a coordinate system with origin at the vertex and x-axis along the parabola s axis of symmetry; 2) find an equation for the parabola; 3) find the depth of the satellite dish at the vertex.
Mathematics in English at HHX Possible Solutions e.g. books and chapters (parts of) References for Mathematics B: References for MathematicsA: Websites: zweigmedia.com, sosmath.com, mathsisfun.com - Haese et al. (2012) Mathematics for the International Student (3rd. Edition). Haese Mathematics Ed. - Wazir et al. (2012) Mathematics Higher Level (2012 Edition). Pearson Edu. Ltd. - Haeussler et al. (2008) Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences (12th Edition) Pearson Edu. Ltd. - Barnett et al. (2015) College Mathematics for Business Economics and Social Sciences (13th Edition) Pearson Edu. Ltd.
Conclusions: Although focus points can be many, it is suggested to: - Work out a progression using a combination of natural and mathematical language Establish a common language let the students communicate (brief presentations) use the mathematics (e.g. repetition) Involve colleges (economics and languages) Involve the managers
Conclusions: - Students can misundertand statements or cannot connect former knowledge with same knowledge in other language (i.e. language barriers) -Finding relevant material is hard and time-consuming -Much in the internet (including Khan) is good for practicing but lacks theory -Most of the results in google are useless so googling for information is hard and time-consuming. - Translation to Danish requirements (bekendtg relsen) is hard and time-consuming
