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Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing?

Franz Lichtenberger

Research Institute for Symbolic Computation (RISC-Linz) Johannes Kepler University Linz, Austria

CME-EI, August 17, 2018

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Status Quo

◮ Several (often highly) intelligent mathematical

software systems are used in K-12 math education.

◮ Support classical engineering mathematics, i.e.,

◮ solving equations, ◮ differentiation and integration, ◮ geometry ◮ . . .

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Status Quo

◮ Several (often highly) intelligent mathematical

software systems are used in K-12 math education.

◮ Support classical engineering mathematics, i.e.,

◮ solving equations, ◮ differentiation and integration, ◮ geometry ◮ . . .

◮ (Hopefully) more enlightenment than incantation!

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Status Quo

◮ Several (often highly) intelligent mathematical

software systems are used in K-12 math education.

◮ Support classical engineering mathematics, i.e.,

◮ solving equations, ◮ differentiation and integration, ◮ geometry ◮ . . .

◮ (Hopefully) more enlightenment than incantation! ◮ Everything seems fine, BUT ...

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

The Times They Are A-Changin’

(Bob Dylan)

◮ Computer Science, especially programming is

gaining importance.

◮ See for example former president Obama’s

“Computer Science For All” – initiative.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

The Times They Are A-Changin’

(Bob Dylan)

◮ Computer Science, especially programming is

gaining importance.

◮ See for example former president Obama’s

“Computer Science For All” – initiative.

◮ Why should this bother us?

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

The Times They Are A-Changin’

(Bob Dylan)

◮ Computer Science, especially programming is

gaining importance.

◮ See for example former president Obama’s

“Computer Science For All” – initiative.

◮ Why should this bother us? ◮ My claim: we are teaching the wrong kind of

mathematics!

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Intelligent vs. Unintelligent Computing

“A program is no more than half a conjecture.” (Edsger W. Dijkstra)

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Intelligent vs. Unintelligent Computing

“A program is no more than half a conjecture.” (Edsger W. Dijkstra)

◮ So, just to write a program is unintelligent computing.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Intelligent vs. Unintelligent Computing

“A program is no more than half a conjecture.” (Edsger W. Dijkstra)

◮ So, just to write a program is unintelligent computing. ◮ For intelligent computing one has to

◮ specify the problem, and then to ◮ prove the full conjecture, i.e.,

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Intelligent vs. Unintelligent Computing

“A program is no more than half a conjecture.” (Edsger W. Dijkstra)

◮ So, just to write a program is unintelligent computing. ◮ For intelligent computing one has to

◮ specify the problem, and then to ◮ prove the full conjecture, i.e.,

◮ prove that the proposed program solves the problem.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Mathematics for Intelligent Computing

◮ Problem Specifications need a language with clear

semantics, i.e., the language of mathematics.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Mathematics for Intelligent Computing

◮ Problem Specifications need a language with clear

semantics, i.e., the language of mathematics.

◮ Proving the conjecture belongs to

the methods of mathematics.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Mathematics for Intelligent Computing

◮ Problem Specifications need a language with clear

semantics, i.e., the language of mathematics.

◮ Proving the conjecture belongs to

the methods of mathematics.

◮ Classical engineering mathematics does not support

intelligent computing!

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Historical Remarks

◮ Around 1980: “Mindstorms” by Seymour Papert ◮ Shows how children can learn all necessary

concepts of programming.

◮ LOGO language and “turtle graphics” regarded as a

contribution to math education.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Historical Remarks

◮ Around 1980: “Mindstorms” by Seymour Papert ◮ Shows how children can learn all necessary

concepts of programming.

◮ LOGO language and “turtle graphics” regarded as a

contribution to math education.

◮ David Stoutemyer (1979):

Teach programming with “Computer Symbolic Math", i.e. computer algebra, in math classes!

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Historical Remarks

◮ Around 1980: “Mindstorms” by Seymour Papert ◮ Shows how children can learn all necessary

concepts of programming.

◮ LOGO language and “turtle graphics” regarded as a

contribution to math education.

◮ David Stoutemyer (1979):

Teach programming with “Computer Symbolic Math", i.e. computer algebra, in math classes!

◮ This was long before personal computers were

available at schools!

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

The “fourth R”

◮ Reading, wRiting, aRithmetik, and progRamming ◮ The basic skills that every child has to learn in

school.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

The “fourth R”

◮ Reading, wRiting, aRithmetik, and progRamming ◮ The basic skills that every child has to learn in

school.

◮ Many educators opposed the separation of

mathematics education from computer science in general, and programming in particular.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Divergence

◮ For a long time I believed that math and CS

education will eventually converge.

◮ The opposite is the case!

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Divergence

◮ For a long time I believed that math and CS

education will eventually converge.

◮ The opposite is the case! ◮ For me this is a sad state because I believe that

◮ mathematical (axiomatic) thinking and ◮ computational (algorithmic) thinking

◮ are just two sides of the same coin,

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Divergence

◮ For a long time I believed that math and CS

education will eventually converge.

◮ The opposite is the case! ◮ For me this is a sad state because I believe that

◮ mathematical (axiomatic) thinking and ◮ computational (algorithmic) thinking

◮ are just two sides of the same coin, ◮ and thus should be taught together!

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

A Blend of Math and CS

◮ The mandatory part:

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

A Blend of Math and CS

◮ The mandatory part: ◮ Ordinary MAthematics, ◮ Programming, ◮ Algorithms and Data Structures

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

A Blend of Math and CS

◮ The mandatory part: ◮ Ordinary MAthematics, ◮ Programming, ◮ Algorithms and Data Structures

(OMA-PADS)

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Ordinary Mathematics

◮ Term used by Leslie Lamport ◮ Not more than logic and sets,

including functions and relations.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Ordinary Mathematics

◮ Term used by Leslie Lamport ◮ Not more than logic and sets,

including functions and relations.

◮ Logic is often regarded as trivial by mathematicians,

not much time allocated.

◮ Should become the “workig language”, much more

effort needed.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Ordinary Mathematics

◮ Term used by Leslie Lamport ◮ Not more than logic and sets,

including functions and relations.

◮ Logic is often regarded as trivial by mathematicians,

not much time allocated.

◮ Should become the “workig language”, much more

effort needed.

◮ Sufficient for writing specifications

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Proving

◮ Usually learned by listening to dozens of proofs in

class.

◮ Mathematicians think that it cannot be taught

explicitely.

◮ CS educators regard rigorous proofs as an overkill.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Proving

◮ Usually learned by listening to dozens of proofs in

class.

◮ Mathematicians think that it cannot be taught

explicitely.

◮ CS educators regard rigorous proofs as an overkill. ◮ For intelligent computing it is a necessity!

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Programming

◮ Belongs to the basic skills. ◮ Every child should learn how to program. ◮ “Computer literacy” or “digital competence” is not

enough!

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Programming

◮ Belongs to the basic skills. ◮ Every child should learn how to program. ◮ “Computer literacy” or “digital competence” is not

enough!

◮ Curriculum proposals available from CS education.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Algorithms and Data Structures

◮ Learn that often several algorithms can solve a given

problem,

◮ some more efficient than others, ◮ clever data structure can reduce time complexity

dramatically.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Algorithms and Data Structures

◮ Learn that often several algorithms can solve a given

problem,

◮ some more efficient than others, ◮ clever data structure can reduce time complexity

dramatically.

◮ Introductory textbooks contain appropriate topics, for

example Sorting and Searching.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Beyond the Mandatory

◮ Mathematics is more than OMA and ◮ Computer Science (Informatics) is more than PADS. ◮ Not easy to decide which topics should remain,

which should be added to new K-12 curricula.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Beyond the Mandatory

◮ Mathematics is more than OMA and ◮ Computer Science (Informatics) is more than PADS. ◮ Not easy to decide which topics should remain,

which should be added to new K-12 curricula.

◮ Computational Thinking and some parts of the blend

can be taught basically “everywhere”.

◮ All these efforts should be supported!

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Beyond the Mandatory

◮ Mathematics is more than OMA and ◮ Computer Science (Informatics) is more than PADS. ◮ Not easy to decide which topics should remain,

which should be added to new K-12 curricula.

◮ Computational Thinking and some parts of the blend

can be taught basically “everywhere”.

◮ All these efforts should be supported! ◮ Difficult as well: At what age should we teach what?

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Implementing OMA-PADS

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Implementing OMA-PADS

◮ Why demand a radical change? ◮ Pressure from industry to teach more of CS,

programming, and computational thinking, but

◮ classical engineering mathematics will not lose

importance.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Implementing OMA-PADS

◮ Why demand a radical change? ◮ Pressure from industry to teach more of CS,

programming, and computational thinking, but

◮ classical engineering mathematics will not lose

importance.

◮ Why not leave math education as it is and ◮ let CS develop?

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Just Add CS to K-12 Curriculum?

◮ CS, including programming, is a demanding field –

comparable to mathematics.

◮ STEM education is very important, but I am against

introducing CS at the cost of the other fields like

◮ languages, humanities, arts, physical education, etc. ◮ (The only(?)) Way out of the dilemma:

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Just Add CS to K-12 Curriculum?

◮ CS, including programming, is a demanding field –

comparable to mathematics.

◮ STEM education is very important, but I am against

introducing CS at the cost of the other fields like

◮ languages, humanities, arts, physical education, etc. ◮ (The only(?)) Way out of the dilemma: ◮ take the hours from math! ◮ More precisely: create a combination of math and

CS, i.e., OMA-PADS.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Experiences at College Level

◮ We have implemented a similar blend in math

courses for CS already in the early 1980s at JKU Linz.

◮ Concept was developed by Bruno Buchberger, I had

the privilege to contribute from the beginning.

◮ Book in German and out of print, pdf copy available

for download.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Experiences at College Level

◮ We have implemented a similar blend in math

courses for CS already in the early 1980s at JKU Linz.

◮ Concept was developed by Bruno Buchberger, I had

the privilege to contribute from the beginning.

◮ Book in German and out of print, pdf copy available

for download.

◮ First semester dedicated to teaching “the language

and methods of mathematics”.

◮ We try to analyze and teach all those aspects of

mathematics that are necessary to treat the whole problem solving process.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Software Tools for Ordinary Mathematics

◮ Proof assistants: an importand topic at the CICM

conference.

◮ Two of them developed at RISC-Linz:

◮ THEOREMA (by B. Buchberger, W. Windsteiger, et

al.)

◮ RISC ProofNavigator (by W. Schreiner)

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Teaching OMA-PADS

◮ Who could teach this combination of math and CS? ◮ Not many math teachers will be willing to take on the

burden of learning the PADS part.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Teaching OMA-PADS

◮ Who could teach this combination of math and CS? ◮ Not many math teachers will be willing to take on the

burden of learning the PADS part.

◮ All teachers have had exposure to mathematics in

their own high school education, thus

◮ it should be easy for CS teachers to teach ordinary

math as well,

◮ besides PADS - which is part of their competence

anyway.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Possible Future Situation

◮ Modeling our physical world with classical

engineering mathematics will remain important, but

◮ there is another kind of math gaining importance for

describing our man-made cyber-physical, web- and cloud-based, mobile and ubiquitous new realities.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Possible Future Situation

◮ Modeling our physical world with classical

engineering mathematics will remain important, but

◮ there is another kind of math gaining importance for

describing our man-made cyber-physical, web- and cloud-based, mobile and ubiquitous new realities.

◮ To learn how to program an app will be regarded

more important than to solve various kinds of equations or to integrate a function.

◮ Mathematical topics will be replaced, piece by piece,

by CS topics.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

A Wake-Up Call

◮ Mathematics educators have to react, otherwise ◮ Computer Science could take over, and

in the worst case

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

A Wake-Up Call

◮ Mathematics educators have to react, otherwise ◮ Computer Science could take over, and

in the worst case

◮ math education as taught in schools today could go

down the river of irrelevance as ancient Latin did some decades ago.

Intelligent Computer Mathematics OR Mathematics for Intelligent Computing? Franz Lichtenberger Motivation Historical Remarks A Blend of Math and CS Implementing OMA-PADS A Wake-Up Call

Thank You!