Mara Alagic IMECT 3 Cambridge July 2002 Technology-Based Representations of Some Mathematical Ideas Mara Alagic Wichita State University Wichita, Kansas,

Mara Alagic IMECT 3 Cambridge July 2002

Technology-BasedRepresentations of Some Mathematical

Ideas

Mara AlagicWichita State University

Wichita, Kansas, USA


Teaching and Learning Mathematics

Standards-based teaching (NCTM, 2000)

Current findings about the nature of human learning (NRC, 2000)

Development of the IC technologiesTechnology Principle (NCTM, 2000)


Mathematics Teaching and Technology

Technology Principle: "Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning" (NCTM, 2000)

Technology in context


Representations

Representing knowledge (Langer, 1989)

Problem-based representationsRepresentation standard (NCTM,

2000)Things before words, concrete

before abstract. (Pestalozzi,1803)


TRANSLATION BETWEEN MODES OF

REPRESENTATION

WRITTEN SYMBOLS

REAL-WORLD EXPERIENCE

MANIPULATIVES = CONCRETE

PICTURES

SPOKEN SYMBOLS

SPOKEN SYMBOLS

PICTURESMANIPULATIVES = CONCRETE

REAL-WORLD EXPERIENCE

WRITTEN SYMBOLS


Conceptual understanding via

multiple representations

Perception-based representationsMultiple representations External and internal

representationsProblem-based representationsTranslations among

representations


MENTAL MODELS/INTERNAL REPRESENTATIONS

EXTERNAL REPRESENTATIONS& TRANSLATIONS


"... the defining characteristic of knowledge workers is that they are themselves changed by the information they process"

Kidd (1994, p. 186)


Questions & Tasks: (Jaelyn & Barbie)

1. DEFINE TESSELLATION(S).

2. ARE THERE DIFFERENT TYPES OF TESSELLATIONS? EXPLAIN YOUR ANSWER.

http://www.mathacademy.com/pr/minitext/escher/index.asp

http://www.coolmath.com/tesspag1.htm


Have Fun Creating Tessellations (Jaelyn & Barbie)

Escher Web Sketch Linkhttp://www-sphys.unil.ch/escher/


Cultural Influences (Jaelyn & Barbie)

IN THE WEB SITES PROVIDED FIND 3 EXAMPLES OF TESSELLATIONS USED IN OTHER CULTURES.

Cultural references http://mathforum.org/sum95/suzanne/historytess.html Cultural tilling exampleshttp://www2.spsu.edu/math/tile/grammar/index.htm


Graphs and SlopeRobin Racette

y = -4x

b = 0; m = -4

-4/1 = down 4, right 1

4/-1 = up 4, left 1

2. y = -1/4x

b = 0; m = -1/4

-1/4 = down 1, right 4

1/-4 = up 1, left 4


More Slope Robin Racette

Try these on your own. y = 1/4x - 4

b = -4, m = 1/4

2. y = -3x

b = 0, m = 3

Graphing y = mx + b on the web


Adam the early bird

Starts at age 22

Invests $3000 for 6 years

At 11% interest

MillionaireMeredith, LuAnne & Dan


Brenda the procrastinatorStarts investing at age 28

Invests $3000 every year thereafter

Until age 63

At 11% interest

MillionaireMeredith, LuAnne & Dan


Who retires with the most? Meredith, LuAnne & Dan

Adam? Brenda?

And how much difference between the final results?

Plan A and B.xls


Meredith, LuAnne, & Dan


Invests $18,000Retires at age 63 with $1,128,249

Invests $108,000Retires at age 63 with $1,265,947

Meredith, LuAnne, & Dan


Graph of Sin and Csc Sarah

Things to Consider• Period?• Amplitude?• Asymptotes?

Differences?• Between function

and its inverse

Green Globs


Sarah


Cos and Sin Wave

A C

time

D

D’

Sin

Cos

Sine/Cosine Tracer

D. Bennett 3/91

Geometer Sketchpad

Sarah


Tangent on a Circle

B

A

time

C

D

Tan

D. Bennett 3/91

Tangent Tracer

Geometer Sketchpad

Sarah


What if I still need help?Ask for helpWebsites

• http://www.drmath.com/• http://www.gomath.com/

Play Green Globs

AND…..

Sarah


TO BECOME AN EXPERT OF TRIG FUNCTIONS

Sarah

Documents

Mara Alagic IMECT 3 Cambridge July 2002 Technology-Based Representations of Some Mathematical Ideas Mara Alagic Wichita State University Wichita, Kansas,