Teaching Math to CS Students – Efforts, Experiences, … · vs MATLAB&SMT or Octave&GinaC Similarities both are CASs (Symbolic Math Toolbox incl.), command line oriented, programmable,

Teach Math

Thomas RisseIIA, HSB

ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions

SAGESimilarities & Differences

Calculus

(Symbolic) Linear Algebra

Numerics

ODEs

Programming

SAGE offers more than MATLAB

Teaching Math to CS Students– Efforts, Experiences, Effects –

Teaching Math Using SAGE

Thomas RisseInstitute for Informatics & Automation, IIAFaculty EEE & CS, Hochschule Bremen

University of Applied [email protected]

Presentation 19.8.2014German Jordanian University

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Agenda1 Professors View

Why is Math Important to CS?

Implications to the Curricula

2 Students View

Learning PreConditions

Observations

3 Projects in the Context of Teaching

4 Math Related Research5 Conclusions

6 SAGE = System for Algebraic & Geometric Experimentation

Similarities & Differences

(Symbolic) Calculus


Numerics

Ordinary Differential Equations

Programming


Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Why is Math Important to CS, toCS Students?

personal answer: one could not do without it . . .

some (optional) modules (in alphabetical order) with somerelevant math

• algorithms and their complexity

• cryptography (discrete math)

• error correcting codes (linear algebra, arithmetic in finite fields)

• generative computer graphics (homogenous coordinates,freeform surfaces: Bézier, B-spline, NURBS etc)

• image processing (discrete Fourier transform etc)

• scientific computing (linear algebra, analysis/calculus)

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Why is Math Important to CS?

my colleagues teach for example

• metrology (stochastics)

• control (Laplace transform)

• modelling and simulation (differential equations)

I also teach computer architecture which links hardware tosoftware issues, i.e. machines to algorithms!

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Implications to the CurriculaWe think math as important as to assign several modules to it:

TI,ISTI,MEI,DSInf MI IFI

1. linear algebra,analytical geome-try

analysis, linearalgebra, analyti-cal geometry

logic, sets, rela-tions, mappings,number systems

2. analysis & ode stochastic, diffe-rential geometry(parametric cur-ves and surfa-ces), numerics

linear algebra,analytical geo-metry, graphs,applied analysis,number theory

3. multidimensionalanalysis, stocha-stic

Math is the only module spanning two or even three semesters.

At the same time: we have to decide

• self contained math modules or

• modules with application specific math integrated

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Students View

http://http:

//www.os-community.de/Gruppe/57555/Mathe_ist_

ein_Arschloch_und_Physik_sein_kleiner_Bruder

http://http://www.os-community.de/Gruppe/57555/Mathe_ist_ein_Arschloch_und_Physik_sein_kleiner_Bruder



Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Learning PreConditions

There are more and more problems [5] like

• dubious competencies acquired in school• extremely vague• very unreliable• insecure knowledge• barely expressible, hardly applicable

• taking notes instead of following the argument

• exercises only for eager beavers . . . ’exercises have nothing to do with exams . . .

• engaging in, examining of, quarrelling with math• entrenchment of math in applications• fluency in dealing with math• sustained learning of math

• dubious time management, procrastination [8],start learning 5min before the exam

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Observations

Growing deficits in basic competencies

• to read and to understand (long) text and formulae

• to write concise answers

• to do mental arithmetic

Growing difficulties

• to observe and to draw conclusions

• to identify (modelling, programming, ...) errors

• to check plausibility

• to model, to abstract, cp OO

The only problem solving strategy is trial and error . . .

Unwillingness to monitor the own learning process!

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Why learn math?

http://www.glasbergen.com/wp-content/gallery/

math-cartoons/toon700.gif

http://www.glasbergen.com/wp-content/gallery/math-cartoons/toon700.gif

http://www.glasbergen.com/wp-content/gallery/math-cartoons/toon700.gif

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


How to learn math?

http://www.glasbergen.com/wp-content/gallery/

math-cartoons/toon-1674.gif

http://www.glasbergen.com/wp-content/gallery/math-cartoons/toon-1674.gif

http://www.glasbergen.com/wp-content/gallery/math-cartoons/toon-1674.gif

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


’in spite of hours of google nosolution found . . . ’

http:

//www.trottermath.net/humor/estimatedtime.gif

http://www.trottermath.net/humor/estimatedtime.gif

http://www.trottermath.net/humor/estimatedtime.gif

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


in the light of [6] no 117

for example

already 1934http://www.openculture.

com/2014/01/

the-president-of-northwestern-university-predicts-online-learning-in-1934.

html

• http://AULIS.

hs-bremen.de

• http://weblearn.

hs-bremen.de/

risse/MAI/docs

• WWW with e.g.http://de.

wikipedia.org

• http://youtube.de

with Jörn’s chanel

• MOOCs likehttp://www.

openculture.com/

math_free_courses

• http://www.

wolframalpha.com

http://www.openculture.com/2014/01/the-president-of-northwestern-university-predicts-online-learning-in-1934.html




http://AULIS.hs-bremen.de

http://AULIS.hs-bremen.de

http://weblearn.hs-bremen.de/risse/MAI/docs



http://de.wikipedia.org

http://de.wikipedia.org

http://youtube.de

http://www.openculture.com/math_free_courses



http://www.wolframalpha.com

http://www.wolframalpha.com

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Teaching Related ProjectsBesides the yearly preparatory courses (WS14/15: with OMB[4]) for the new students

1 integration of text and algorithm: documents whichcompute, see [6] no 53,55,76

• numerik.pdf, numerics.pdf: basic numericalgorithms, CORDIC ([6] no 66,67,68)

• puzzles.pdf, ePuzzles.pdf: some riddles, primenumbers, finite fields GF(pm), cryptography (DES, AES,RSA, ECC), compression, intuition and probability

• heath.pdf: addons to M.T.Heath: Scientific Computing

2 Känguru competitions in 2001–2004 and collections ofKänguru problems with solutions since 1998 http:

//www.weblearn.hs-bremen.de/risse/MAI/docs/Kanguru/

3 televise – a notebook university project, see [6] no 59–65• database of exercises⇒ LOSPN, see [6] no 69,71,72• tutors provide online support when students work on

assignments

http://www.weblearn.hs-bremen.de/risse/MAI/docs/Kanguru/

http://www.weblearn.hs-bremen.de/risse/MAI/docs/Kanguru/

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Teaching Related Projects’

4 self organized learning, cp [3], see [6], no 82,85,86

5 graphical inputs in exams, e.g. sketch a normal parabola,see [6] no 74,75,77,78

6 Fit in Math, FiM, uses CAS• to pose questions on ILIAS• to evaluate the answers of the learners• to provide feedback

in drill/self-test/exam mode, see [6] no 88,93,103,106

7 learning math3 in projects, SS12: Systems of Springsand WS12/13: Baumgartner up/down, many bodies,see [6] no 120,124,125

8 collaborative coaching, WS13/14, see [6] no 131• weekly obligatory homework• support when working on homework• relate/integrate math3 and metrology

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Teaching Related Projects”

9 flipped classroom in physics, SS13, SS14, cp [3], [2]10 reflections on observations of the learning behaviour of

my students• pocket calculators, see [6] no 102,107• first year students attitude, see [6] no 121• students being online all the time, see [6] no 117

11 orientation• How much numerics is needed? see [6] no 70• Math in relation to Science: Body and Soul framework,

see [6] no 95,98

12 political demands• pocket calculators, see [6] no 90,104,105• advisory opinion, petition, see http:

//www.weblearn.hs-bremen.de/risse/papers/GTR/

http://www.weblearn.hs-bremen.de/risse/papers/GTR/

http://www.weblearn.hs-bremen.de/risse/papers/GTR/

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Math Related Research

• eLearning, eTests, see [6] no 49–52,56–58,74,75,77,78,FiM 87,88,93,103,106

• CORDIC algorithms, see [6] no 66,67,68

• The Non-Intuitive in Mathematics, see [6] no 79,80,81

• Elliptic Curve Cryptography, ECC, see [6] no 83,84,89

• Markov chains, see [6] no 91,92,94

• Zernike polynomials, see [6] no 119

• SAGE, see [6] no 108,110–116,118,122

• Goppa Codes, see [6] no 109,115,116,126

• Random Numbers, see [6] no 127,130,132

• System for Teaching and Assessment using a Computeralgebra Kernel, STACK, http://www.stack.bham.ac.uk/,see FiM and [6] no 123,128,134

http://www.stack.bham.ac.uk/

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Some Concluding Remarks

• growingly firm conviction of the importance of math forCS⇒ no progress without mathI’m not alone: just google ’math computer science’:the first hits are from Princeton (LL), MIT (open courseware: discrete math, algorithms), Oxford (stressinginterdependence), Imperial College (’jointly’), Yale(interdepartmental major), CalTech (Computing +Mathematical Sciences) . . .

• growingly firm conviction that students learn more if theyare asked to solve problems from all parts of science likee.g. those inSteven C. Chapra: Applied Numerical Methodsusing some CAS, e.g. the open source SAGE, Systemfor Algebraic and Geometric Experimentation.

• if one really wants to integrate research and teaching(cp my research) then students need even more math . . .

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


SAGE = System for Algebraic and Geometric Experimentation

vs MATLAB&SMT or Octave&GinaCSimilarities

both are CASs (Symbolic Math Toolbox incl.), command lineoriented, programmable, problem oriented, extendable etc.both feature graphical capabilities, different workspaces,interpreted, precompiled, object orientation etc.

DifferencesSAGE MATLABlocal implementation local implementation& web interfacePython m-filesmany predefined data types few predefined data typesobjects handlesmodules toolboxesopen source proprietary/open sourcebrowser installation

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


(Symbolic) Calculus

visualisationplot(sin(x)) ezplot('sin(x)')

plot(sin(x),-3,3) ezplot('sin(x)',-3,3)

p=plot(sin(x),-3,3, \color='red');

q=plot(cos(x),-3,3, \color='blue');

show(p+q);

hdl=ezplot('sin(x)',-3,3);

set(hdl,'color','r');

hold on;

hdl=ezplot('cos(x)',-3,3);

set(hdl,'color','b');

hold off;

plot(1/x,detect_poles=True) ezplot('1/x',-1,1)

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


(Symbolic) Calculus

Differentiation and Integration

diff(sin(x)/x,x) diff(sin(x)/x,x)

integral(x*sin(x),x) int(x*sin(x),x)

integral(exp(-x∧2/2),x,0,1) int(exp(-x∧2/2),x,0,1)

of course, with numerical evaluation, e.g.

N(integral(sin(x)/x,x,0,1)) N(int(sin(x)/x,x,0,1))

e.g. one of my GJU students asked for power and Taylor series

f.taylor(x,xo,ord) taylor(f,x,xo,...)

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Solving equations symbolically

e.g. linear (systems)

vrs=var('a b c d e f x y');

solve([ \a*x+b*y==c, \d*x+e*y==f],x,y)

syms a b c d e f x y;

[x y] = solve( ...

'a*x+b*y-c', ...

'd*x+e*y-f',x,y)

e.g. non-linear

vars=var('a b c d x');

solve( \a*x∧3+b*x∧2+c*x+d==0,x)

syms a b c d x;

solve( ...

'a*x∧3+b*x∧2+c*x+d')

e.g. differential

t = var('t');

x = function('x',t);

desolve(diff(x,t)+x==1,[x,t])

% t is independent

% var by default

dsolve('Dx+x=1')

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Linear Algebra

vectors & matricespnts = [(random(),random())\for _ in range(5)];

p = line(pnts);

c = cos(pi/4);

s = sin(pi/4);

R=N(matrix([[c,s],[-s,c]]));

pnts = matrix(pnts)*R;

q=line([(pnts[i,0],pnts[i,1])\for i in range(5)],\color='red');

(p+q).show()

plgn = rand(2,5);

plot(plgn(1,:), ...

plgn(2,:));

hold on;

c=cos(pi/4);

s=sin(pi/4);

R = [c s;-s c];

plgn = R*plgn;

plot(plgn(1,:), ...

plgn(2,:),'r');

hold off;

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming



linear equations

vars = var('a,b,c,d,e,f');

A = matrix([[a,b],[c,d]]);

b = vector([e,f]); A\b

syms a b c d e f;

A = [a b;c d];

b = [e;f]; A\b

or again, also symbolically

A = matrix([[1,2,3], \[3,2,1],[1,1,1]]);

b =vector([0,-4,-1]); A\b

A = sym([1,2,3; ...

3,2,1;1,1,1]);

b =sym([0;-4;-1]); A\b

eigenvalues & eigenvectors

A = matrix([[1,2,3], \[3,2,1],[1,1,1]]);

A.eigenvalues(); # or

A.eigenvectors_right();

A = sym([1,2,3; ...

3,2,1;1,1,1]);

eig(A) % or

[V,D] = eig(A);

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Numerics

both CASs based on based on BLAS or LAPACK routines

• solving linear equations by decomposition:LU, Cholesky, Schur, SVD

• least squares, eigenvalues & eigenvectors

• solving non-linear equations

• quadrature, double/tripple quadrature

• solving ordinary differential equations

x = var('x');

find_root( \cos(x)==sin(x),0,pi/2)

% x0 is start guess

% or start intervall

fzero( ...

@(x)(cos(x)-sin(x)), x0)

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Ordinary Differential Equations – IVP

Volterra-Lottka IVP model y′ =(

y1(a− by2)y2(−c + dy1)

)for

t ∈ [0, 25] and for two populations y1(t) and y2(t).

# initialize

# a,b,c,d,y0[0],y0[1]

def odefun(y,t):

return [ \y[0]*(a-b*y[1]), \y[1]*(-c+d*y[0])];

t = srange(0,25,0.01);

y = odeint(odefun,y0,t)

% initialize

% a,b,c,d,y0(1),y0(2)

function fvl = odef(t,y)

fvl=[y(1)*(a-b*y(2));...

y(2)*(-c+d*y(1))];

end

tspan = [0 25];

[t y_ode45] = ...

ode45(@odef,tspan,y0);

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Ordinary Differential Equations – BVP

Some BVP u′′ + 2u′ + u = 0 for 0 ≤ x ≤ π2 with

u(0) = 3, u(π2 ) = 2 is equivalent to y′ =(

y2

−2y2 − y1

)vars = var('x');

de = diff(y,x,2)+ \2*diff(y,x)+y==0;

sln = desolve(de, \y,[0,3,pi/2,2]);

plot(sln(x),(x,0,pi/2));

# there are packages

# like the numerical

# scikits.bvp_solver

# also ...

f = @(x,y) [y(2); ...

-y(2)-y(1)];

g = @(ya,yb)[ya(1)-3; ...

yb(1)-2];

x = linspace(0,pi/2,10)';

yguess=@(x)[3-2*x/pi; ...

-2/pi];

init = bvpinit(x,yguess);

sln = bvp4c(f,g,init);

xp = 0:0.01:pi/2;

plot(xp,deval(sln,xp,1));

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Animations

anmt = animate( \[point([ \2*sin(pi/8*cos(t)),\2*(1-cos(pi/8*cos(t)))])\for t in \srange(0,11,0.1)],\xmin=-1,xmax=1, \ymin=0,ymax=2)

anmt.show()

plot([-2,2,0,0], ...

[0,0,0,2],'k');

hold on;

h=plot(0,0,'.k');

while true

set(h,'xData', ...

2*sin(pi/8*cos(t)));

set(h,'yData', ...

2-2*cos(pi/8*cos(t)));

drawnow;

end; hold off;

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


Interactions

@interact

def NN(a=(0,2), \b=1, c=[0,2,4])

p=plot( \a*x∧2+b*x+c,\

-2,2);

p.show();

% begin code snippet

a = get(ha,'Value');

b=double(get(hb,'String'));

c = get(hc,'Value');

f = @(x) a*x.∧2+b*x+c;ezplot(f);

% end code snippet

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming



in the face of a reference manualhttp://www.sagemath.org/doc/reference/

with more than 8000 pages I can give some examples only:

• Discrete Mathematics, e.g.• arithmetic in any finite fields, cp my error correcting codes• partitions, cp Ala’a problem• graph theory, cp my scheduling problems• combinatorics

• Rings, Fields, Algebras

• Groups, Monoids, Matrices, Modules

• e.g. Combinatorial Geometry, Cell Complexes and theirHomology, Differential Forms, Parametrized Surfaces

• Number Theory, Algebraic Geometry, cp my ECC

• . . .

http://www.sagemath.org/doc/reference/

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


My Personal Outlook

http://www.sciencecartoonsplus.com/gallery/

math/math155_public-spirited-mathematician.gif

http://www.sciencecartoonsplus.com/gallery/math/math155_public-spirited-mathematician.gif

http://www.sciencecartoonsplus.com/gallery/math/math155_public-spirited-mathematician.gif

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


References

[1] Sidney Harris: Math Cartoons;http://www.sciencecartoonsplus.com/gallery/math/index.php

http://www.sciencecartoonsplus.com/gallery/math/

math115_you-cant-be-a-mathematician.gif

[2] Jörn Loviscach: Didactics and Technology;http://www.j3l7h.de/videotech.html

http://www.sciencecartoonsplus.com/gallery/math/index.php

http://www.sciencecartoonsplus.com/gallery/math/math115_you-cant-be-a-mathematician.gif

http://www.sciencecartoonsplus.com/gallery/math/math115_you-cant-be-a-mathematician.gif

http://www.j3l7h.de/videotech.html

Teach Math


ProfessorsWhy Math

Implications

StudentsConditions

Observations

Projects

Research

Conclusions


Calculus


Numerics

ODEs

Programming


References’

[3] Jörn Loviscach: Das Inverted Classroom Model;https://www.youtube.com/watch?v=Ob8yyXf2tOU

[4] MUMIE – Multimediale Mathematikausbildung: Online MathematikBrückenkurs; https://www.om-bridge.de/

[5] Ben Orlin: What It Feels Like to Be Bad at Math – My hazy, anxious,defensive procrastination made me a better teacher.http://www.slate.com/articles/health_and_science/science/

2013/04/math_teacher_explains_math_anxiety_and_

defensiveness_it_hurts_to_feel_stupid.single.html

[6] Thomas Risse: papershttp://www.weblearn.hs-bremen.de/risse/papers/papers.htm

[7] William Stein et al: SAGE www.sagemath.org

[8] David Weigert: The Day Before;http://davidweigert.com/The-Day-Before

https://www.youtube.com/watch?v=Ob8yyXf2tOU

https://www.om-bridge.de/

http://www.slate.com/articles/health_and_science/science/2013/04/math_teacher_explains_math_anxiety_and_defensiveness_it_hurts_to_feel_stupid.single.html



http://www.weblearn.hs-bremen.de/risse/papers/papers.htm

www.sagemath.org

http://davidweigert.com/The-Day-Before

Teaching Math to CS Students – Efforts, Experiences, … · vs MATLAB&SMT or Octave&GinaC Similarities both are CASs (Symbolic Math Toolbox incl.), command line oriented, programmable,

Documents