Computer-supported Mathematics Learning: What do we really want? - - PowerPoint PPT Presentation

▶
computer supported mathematics learning what do we really
SMART_READER_LITE
LIVE PREVIEW

Computer-supported Mathematics Learning: What do we really want? - - PowerPoint PPT Presentation

Mar 09, 2023 288 likes •1.26k views

Computer-supported Mathematics Learning: What do we really want? Prof. Ulrich Kortenkamp, University of Education Karlsruhe TU Eindhoven, June 6-7, 2011 TELMME Conference on E-Learning and Mathematics

slide-1
SLIDE 1

Computer-supported Mathematics Learning: What do we really want?

  • Prof. Ulrich Kortenkamp, University of Education Karlsruhe

TU Eindhoven, June 6-7, 2011 TELMME Conference on E-Learning and Mathematics

  • Montag, 6. Juni 2011
slide-2
SLIDE 2

source: xkcd.com

Montag, 6. Juni 2011

slide-3
SLIDE 3

Moore's Law

Montag, 6. Juni 2011

slide-4
SLIDE 4

Moore's Law

(Wrong, but more spectacular version)

Montag, 6. Juni 2011

slide-5
SLIDE 5

Moore's Law

(Wrong, but more spectacular version)

Computing power doubles every 18 months.

Montag, 6. Juni 2011

slide-6
SLIDE 6

Moore's Law

(Wrong, but more spectacular version)

Computing power doubles every 18 months. Or: Every two years everything becomes twice as good

Montag, 6. Juni 2011

slide-7
SLIDE 7

Montag, 6. Juni 2011

slide-8
SLIDE 8

Year

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

Montag, 6. Juni 2011

slide-9
SLIDE 9

Year Harddisk

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 60.000.000 60.000.000.000 120.000.000.000 320.000.000.000 1.000.000.000.000 4.000.000.000.000

Montag, 6. Juni 2011

slide-10
SLIDE 10

Year Harddisk iPod

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 60.000.000 60.000.000.000 120.000.000.000 5.000.000.000 320.000.000.000 16.000.000.000 1.000.000.000.000 32.000.000.000 4.000.000.000.000

Montag, 6. Juni 2011

slide-11
SLIDE 11

Year Harddisk iPod MHz

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 4 60.000.000 25 60.000.000.000 450 120.000.000.000 5.000.000.000 800 1.670 320.000.000.000 16.000.000.000 3.660 1.000.000.000.000 32.000.000.000 4.800 4.000.000.000.000

Montag, 6. Juni 2011

slide-12
SLIDE 12

Year Harddisk iPod MHz Memory

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 4 16.384 60.000.000 25 1.048.576 60.000.000.000 450 536.870.912 120.000.000.000 5.000.000.000 800 1.073.741.824 1.670 320.000.000.000 16.000.000.000 3.660 3.221.225.472 1.000.000.000.000 32.000.000.000 4.800 4.000.000.000.000 4.294.967.296

Montag, 6. Juni 2011

slide-13
SLIDE 13

Year Harddisk iPod MHz Memory E-Mail

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 4 16.384 60.000.000 25 1.048.576 432 465 400 868 1.472 60.000.000.000 450 536.870.912 1.364 2.007 120.000.000.000 5.000.000.000 4.667 800 1.073.741.824 2.762 4.250 1.670 8.621 13.271 320.000.000.000 16.000.000.000 3.660 14.780 3.221.225.472 17.684 1.000.000.000.000 32.000.000.000 4.800 20.069 21.065 4.000.000.000.000 4.294.967.296

Montag, 6. Juni 2011

slide-14
SLIDE 14

1,00 10,00 100,00 1000,00 10000,00 100000,00 1000000,00 10000000,00 100000000,00 1000000000,00 10000000000,00 1983 1990 1995 2000 2005 2010

On a log scale linear extrapolation seems to be correct Harddisk iPod MHz RAM email

Montag, 6. Juni 2011

slide-15
SLIDE 15

1,00 10,00 100,00 1000,00 10000,00 100000,00 1000000,00 10000000,00 100000000,00 1000000000,00 10000000000,00 1983 1990 1995 2000 2005 2010

On a log scale linear extrapolation seems to be correct Doubling every two years… Harddisk iPod MHz RAM email

Montag, 6. Juni 2011

slide-16
SLIDE 16

We cannot deny this development.

Montag, 6. Juni 2011

slide-17
SLIDE 17

Small but powerful computers will be standard

Montag, 6. Juni 2011

slide-18
SLIDE 18

Small but powerful computers will be standard

(a) Mobile phone: only 10 years "behind" laptop (b) Children will use parents' old phones (a) & (b) => In 2030 every student in the class will have such a device

Montag, 6. Juni 2011

slide-19
SLIDE 19

This will change the teaching and learning of mathematics.

(a) & (b) => In 2030 every student in the class will have such a device

Montag, 6. Juni 2011

slide-20
SLIDE 20

Tools and Content.

Montag, 6. Juni 2011

slide-21
SLIDE 21

Facts and Trends

Tools and Content. ①

Montag, 6. Juni 2011

slide-22
SLIDE 22

Montag, 6. Juni 2011

slide-23
SLIDE 23

What ICT tools do we have?

Montag, 6. Juni 2011

slide-24
SLIDE 24

What ICT tools do we have?

DGS

Dynamic Geometry Software

CAS

Computer Algebra Systems

EXCEL

Spreadsheet Applications

Office

Word, Powerpoint, …

Internet

Browsing, E-Mail, …

Web2.0

Social Networks, Wiki, Collaboration

3D

Virtual Reality, Visualization

Games

Edutainment, FPS, …

DB

Data Bases

Notes

Mindmaps, ToDo, …

LMS

Learning Managment Systems (Moodle)

Fathom

and others

Labs

Coach, Mindstorms, …

Logo

and Programming

Montag, 6. Juni 2011

slide-25
SLIDE 25

DGS

Dynamic Geometry Software

CAS

Computer Algebra Systems

EXCEL

Spreadsheet Applications

Office

Word, Powerpoint, …

Internet

Browsing, E-Mail, …

Web2.0

Social Networks, Wiki, Collaboration

Games

Edutainment, FPS, …

Notes

Mindmaps, ToDo, …

LMS

Learning Managment Systems (Moodle)

@home @school @math

Montag, 6. Juni 2011

slide-26
SLIDE 26

Fact: The Tools are powerful

Montag, 6. Juni 2011

slide-27
SLIDE 27

Trend: Tool Integration

Montag, 6. Juni 2011

slide-28
SLIDE 28

Trend: Tool Integration

DGS

Dynamic Geometry Software

CAS

Computer Algebra Systems

EXCEL

Spreadsheet Applications

3D

Virtual Reality, Visualization

Fathom

and others

Labs

Coach, Mindstorms, …

Logo

and Programming

Montag, 6. Juni 2011

slide-29
SLIDE 29

Trend: Content

Montag, 6. Juni 2011

slide-30
SLIDE 30

Find content at i2geo.net

Montag, 6. Juni 2011

slide-31
SLIDE 31

Tools and Content.

Montag, 6. Juni 2011

slide-32
SLIDE 32

Tools are powerful Content is available Find it at i2geo.net

Tools and Content.

Montag, 6. Juni 2011

slide-33
SLIDE 33

Setting the stage.

Montag, 6. Juni 2011

slide-34
SLIDE 34

Instrumentation, Orchestration and more

Setting the stage. ②

Montag, 6. Juni 2011

slide-35
SLIDE 35

Is ICT-based teaching indeed better?

Montag, 6. Juni 2011

slide-36
SLIDE 36

No.

Is ICT-based teaching indeed better?

Montag, 6. Juni 2011

slide-37
SLIDE 37

Is ICT-based teaching indeed better?

Wrong question.

Montag, 6. Juni 2011

slide-38
SLIDE 38

How can we use technology in

  • rder to improve teaching and

learning?

Montag, 6. Juni 2011

slide-39
SLIDE 39

Curriculum says: We use it to save time.

Montag, 6. Juni 2011

slide-40
SLIDE 40

Is this a good tool?

Montag, 6. Juni 2011

slide-41
SLIDE 41

Is this a good tool?

Montag, 6. Juni 2011

slide-42
SLIDE 42

I am using … … for … … in order to … … as a …

Montag, 6. Juni 2011

slide-43
SLIDE 43

Instrumental Genesis: An artefact becomes an instrument Both the mental schemes and techniques for using it and the mathematical understanding embedded into it are developed. Michèle Artigue

Montag, 6. Juni 2011

slide-44
SLIDE 44

Instrumental Genesis: An artefact becomes an instrument While learning how to use a hammer the students understand the deeper meaning of putting things together. Michèle Artigue

Montag, 6. Juni 2011

slide-45
SLIDE 45

How to –that is, what for– use a tool

  • is easy to describe for ordinary tools
  • is much harder for digital tools

Even for content it is difficult to define the situation

  • leads to huge theories and description

languages: LOM, IMS LD, …

DGS

Dynamic Geometry Software

CAS

Computer Algebra Systems

EXCEL

Spreadsheet Applications

3D

Virtual Reality, Visualization

Fathom

and others

Labs

Coach, Mindstorms, …

Logo

and Programming

Montag, 6. Juni 2011

slide-46
SLIDE 46

Trouche: Instrumental Orchestration (2004)

Montag, 6. Juni 2011

slide-47
SLIDE 47

1.Choose the instruments

create a didactical configuration that describes the artefacts (tools and content)

2.Create a musical score for each instrument

describe how you and the students are going to exploit the didactical configuration (exploitation mode)

3.Perform (Drijvers et al, 2010)

Include the ad hoc decisions made, take into account what is really happening in the didactical performance

Montag, 6. Juni 2011

slide-48
SLIDE 48

Thelonius Monk,

  • ca. 1947

Photo: William P. Gottlieb

Montag, 6. Juni 2011

slide-49
SLIDE 49

Intergeo: We must not only put this in the metadata of learning objects, but we must use this structural approach for continuous professional development (CPD) of teachers Consequence: We must find theories that can be handled by the users and the content providers or developers

Montag, 6. Juni 2011

slide-50
SLIDE 50

Setting the stage.

Montag, 6. Juni 2011

slide-51
SLIDE 51

There are more than enough theories and formalizations that describe learning situations

Setting the stage. ②

Montag, 6. Juni 2011

slide-52
SLIDE 52

Evolution.

Montag, 6. Juni 2011

slide-53
SLIDE 53

How to create: Design Patterns, User Stories and more

Evolution. ③

Montag, 6. Juni 2011

slide-54
SLIDE 54

Erich Gamma, Richard Helm, Ralph Johnson, and John Vlissides. Design Patterns: Elements of Reusable Object-Oriented Software. Addison-Wesley, Reading, MA, 1995.

Design Patterns

Montag, 6. Juni 2011

slide-55
SLIDE 55
  • formalized description
  • answer to a problem, goal or situation
  • several examples available
  • varying levels and granularity
  • provide a common language for standard

situations Examples: metavariation, supplantation, linked representations, referee, funnel & see tomorrow

Design Patterns

Alexander

Montag, 6. Juni 2011

slide-56
SLIDE 56

Metavariation Pattern

  • Summary:

Students can change a situation as a whole to explore special cases and generalities

  • Problem/Situation:

A mathematical notion, object or theorem shall be understood by

  • students. Students lack experience with the object as a whole and

see only isolated cases.

  • Solution:

Use an interactive visualization where students can vary a situation as a whole. Encourage them to create border cases and to discuss the dynamic behavior of the whole object under modifications

  • Examples:

p-norm, triangle problem(Hoffkamp)

Montag, 6. Juni 2011

slide-57
SLIDE 57

Linked Representations Pattern

  • Summary:

Students understand a concept better by varying between different representations

  • Problem/Situation:

Students use only one representation of a mathematical concept and cannot connect one representation (say, a graph of a function) with other representations (the function term, a table). […]

  • Solution:

Provide an interactive visualization of several representations where students can change either representation and all others are changed accordingly.

  • Examples:

changing parabola parameters, primal-dual view of the projective plane, linked symbolic and graphic representation of simple addition

  • n a multi-touch table

Montag, 6. Juni 2011

slide-58
SLIDE 58

Referee Pattern

  • Summary:

Students are engaging in discussions about math and should be able to resolve conflicts that might arise.

  • Problem/Situation:

The Students should discuss about mathematics, but do not know enough to be able to resolve conflicts on their own. The teacher is not available to help.

  • Solution:

Provide a software tool that can answer “right or wrong”-questions about the current topic.

  • Examples:

using a calculator for checking arithmetic, using Visage to check whether a graph is Eulerian or not.

Montag, 6. Juni 2011

slide-59
SLIDE 59

Funnel Antipattern

  • Summary:

The teacher is asking questions in a way that students cannot go wrong and will funnel to the correct answer.

  • Problem/Situation:

With a certain expectation, a teacher is tempted to formulate questions in a way that the students can guess the answers. Students can master this without any deeper understanding.

  • Solution(s):

Stop asking specific questions, but encourage students to ask

  • themselves. – Ask simple open questions like ”why?” – offer the

discussion lead to a student who does not know the answer – …

  • Examples:

Too many to mention…

Montag, 6. Juni 2011

slide-60
SLIDE 60

User Stories

Montag, 6. Juni 2011

slide-61
SLIDE 61
  • A short story that tells from a users’

perspective what (s)he expects

  • Describes requirements to a tool or content

(or orchestration) in an informal way

  • Should be written and refined by users

(learners, teachers) and developers together

  • Epic stories will be decomposed into short

stories.

  • Serve to initiate a discussion

User Stories

Montag, 6. Juni 2011

slide-62
SLIDE 62

Epic user story for whiteboard teaching

Anna, a primary school teacher, wants to introduce subtraction of three-digit numbers in grade 3. She knows that some children have difficulties with the decimal system. During her lesson, she realizes that she would like to help the children to visualize the “borrowing of a ten“. She opens the virtual tool chest and takes

  • ut the decimal system table. After placing some

tokens on the table by tapping on the board, she moves a token from the tens to the ones. The token will become ten tokens. To show that the represented number did not change, she moves back and forth between tens and ones. The table does the counting of tokens automagically. […]

Montag, 6. Juni 2011

slide-63
SLIDE 63

User story for borrowing

As a teacher, I want to be able to borrow from the tens or hundreds and see the result.

Montag, 6. Juni 2011

slide-64
SLIDE 64

Refactoring

Montag, 6. Juni 2011

slide-65
SLIDE 65

In programming, we can:

  • rename variables with correct scope
  • extract methods
  • introduce variables
  • identify common code
  • …

We want this for mathematics!

Refactoring

Montag, 6. Juni 2011

slide-66
SLIDE 66

My User Stories:

  • As a researcher, I want to be able extract a

lemma from a scientific paper and re-use

  • that. If similar cases can be handled by the

same lemma, my text will be reformulated acoordingly.

  • As a referee, I want papers to come with

certificates that show me the correctness of a lemma at least for some special cases.

Refactoring

Montag, 6. Juni 2011

slide-67
SLIDE 67

A mathematician’s IDE

Montag, 6. Juni 2011

slide-68
SLIDE 68

Can we be sure that such reformulation won‘t change the correctness?

Montag, 6. Juni 2011

slide-69
SLIDE 69

Math Unit Testing

Montag, 6. Juni 2011

slide-70
SLIDE 70

Montag, 6. Juni 2011

slide-71
SLIDE 71

Montag, 6. Juni 2011

slide-72
SLIDE 72

Basic Idea

  • Use the computer for verification instead
  • f finding a solution
  • Write tests in CAS before you try to find

the solution

  • Solution finding is next step
  • The task is solved when all tests are

fulfilled and you cannot think of new ones

Math Unit Testing

Montag, 6. Juni 2011

slide-73
SLIDE 73

Formulate tests with the computer Find a solution Check the solution automatically

Can you come up with more tests?

Done! Task Test successful?

No Yes Yes No

Math Unit Testing

Montag, 6. Juni 2011

slide-74
SLIDE 74

Montag, 6. Juni 2011

slide-75
SLIDE 75

Example exercise 526 - 35 = ?

Montag, 6. Juni 2011

slide-76
SLIDE 76

Example exercise 526 - 35 = ? Tests ?<500

Montag, 6. Juni 2011

slide-77
SLIDE 77

Example exercise 526 - 35 = ? Tests ?<500 ?>400

Montag, 6. Juni 2011

slide-78
SLIDE 78

Example exercise 526 - 35 = ? Tests ?<500 ?>400 ?+35 = 526

Montag, 6. Juni 2011

slide-79
SLIDE 79

Example exercise 526 - 35 = ? Tests ?<500 ?>400 ?+35 = 526 ?+9 = 500

Montag, 6. Juni 2011

slide-80
SLIDE 80

Example exercise 526 - 35 = ? Tests ?<500 ?>400 ?+35 = 526 ?+9 = 500 Solution

Montag, 6. Juni 2011

slide-81
SLIDE 81

Example exercise 526 - 35 = ? Tests ?<500 ?>400 ?+35 = 526 ?+9 = 500 Solution ? = 491

Montag, 6. Juni 2011

slide-82
SLIDE 82

Example exercise: Find the intersection of y = 2x+3 and y = 8-3x!

Montag, 6. Juni 2011

slide-83
SLIDE 83

Solution of the exercise is trivial with CAS using solve But we want students to understand that we are talking about a point that must lie on both lines defined by the equations.

Montag, 6. Juni 2011

slide-84
SLIDE 84

Find the intersection of y = 2x+3 and y = 8-3x!

Montag, 6. Juni 2011

slide-85
SLIDE 85

Write tests for the following statements

Find the intersection of y = 2x+3 and y = 8-3x!

Montag, 6. Juni 2011

slide-86
SLIDE 86

Write tests for the following statements

  • (x;y) = (4;11) is not a solution

Find the intersection of y = 2x+3 and y = 8-3x!

Montag, 6. Juni 2011

slide-87
SLIDE 87

Write tests for the following statements

  • (x;y) = (4;11) is not a solution
  • (x;y) = (2;2) is not a solution

Find the intersection of y = 2x+3 and y = 8-3x!

Montag, 6. Juni 2011

slide-88
SLIDE 88

Write tests for the following statements

  • (x;y) = (4;11) is not a solution
  • (x;y) = (2;2) is not a solution
  • (x;y) ist not a solution

Find the intersection of y = 2x+3 and y = 8-3x!

Montag, 6. Juni 2011

slide-89
SLIDE 89

Evolution.

Montag, 6. Juni 2011

slide-90
SLIDE 90

There are ways to know what we want.

Evolution. ③

Montag, 6. Juni 2011

slide-91
SLIDE 91

Finally.

Montag, 6. Juni 2011

slide-92
SLIDE 92

Shape 2030! Remember: Math first.

Finally.

Montag, 6. Juni 2011

slide-93
SLIDE 93

We use ICT …

  • to enable students to build models
  • to enable students to run simulations
  • to connect real-world data to simulations
  • to visualize data
  • to explore abstract concepts interactively
  • to pose questions and challenge students
  • to support the development of Grundvorstellungen
  • to enable students to argue about mathematics
  • to guide students both explicitely and subtle
  • to (self-)assess students' solutions and give feedback
  • to transfer between different representations
  • to find out about the students concepts (diagnose)
  • …

Montag, 6. Juni 2011

slide-94
SLIDE 94

Tell me your stories!

I‘m happy to listen to user stories for

  • 3D geometry software
  • multi-touch enabled DGS
  • Integrated math environments
  • online repositories for interactive content
  • … anything else related to

E-learning and Mathematics kortenkamp@cermat.org

  • Montag, 6. Juni 2011
slide-95
SLIDE 95

joint work with Chris Dohrmann, PH Karlsruhe

Montag, 6. Juni 2011

slide-96
SLIDE 96

joint work with Chris Dohrmann, PH Karlsruhe

Montag, 6. Juni 2011