Transcript Technology-Based Representations of Some Mathematical Ideas Mara Alagic

Technology-Based
Representations of Some
Mathematical Ideas
Mara Alagic
Wichita State University
Wichita, Kansas, USA
Mara Alagic
Cambridge
IMECT 3
July
2002
Teaching and Learning
Mathematics
Standards-based
teaching (NCTM,
2000)
Current findings about the nature of
human learning (NRC, 2000)
Development of the IC technologies
Technology Principle (NCTM, 2000)
Mara Alagic
Cambridge
IMECT 3
July
2002
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
Mara Alagic
Cambridge
IMECT 3
July
2002
Representations
Representing
knowledge (Langer, 1989)
Problem-based representations
Representation standard (NCTM,
2000)
Things
before words, concrete before abstract.
(Pestalozzi,1803)
Mara Alagic
Cambridge
IMECT 3
July
2002
MANIPULATIVES
= CONCRETE
MANIPULATIVES
= CONCRETE
PICTURES
TRANSLATION
BETWEEN MODES OF
REPRESENTATION
REAL-WORLD
REAL-WORLDEXPERIENCE
EXPERIENCE
WRITTEN
SYMBOLS
WRITTEN
SYMBOLS
Mara Alagic
Cambridge
IMECT 3
July
2002
PICTURES
SPOKEN
SYMBOLS
SPOKEN
SYMBOLS
Conceptual understanding via
multiple representations
Perception-based
representations
Multiple representations
External and internal representations
Problem-based representations
Translations among representations
Mara Alagic
Cambridge
IMECT 3
July
2002
MENTAL MODELS/
INTERNAL REPRESENTATIONS
EXTERNAL
REPRESENTATIONS
& TRANSLATIONS
Mara Alagic
Cambridge
IMECT 3
July
2002
"... the defining characteristic of
knowledge workers is that they
are themselves changed by the
information they process"
Kidd (1994, p. 186)
Mara Alagic
Cambridge
IMECT 3
July
2002
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
Mara Alagic
Cambridge
IMECT 3
July
2002
Have Fun Creating
Tessellations
(Jaelyn & Barbie)
Escher Web Sketch Link
http://www-sphys.unil.ch/escher/
Mara Alagic
Cambridge
IMECT 3
July
2002
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 examples
http://www2.spsu.edu/math/tile/grammar/index.htm
Mara Alagic
Cambridge
IMECT 3
July
2002
Graphs and Slope
Robin 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
Mara Alagic
Cambridge
IMECT 3
July
2002
More Slope
Graphing y = mx + b on the web
Robin Racette
Try these on your own.
 y = 1/4x - 4
b = -4, m = 1/4
2. y = -3x
b = 0, m = 3
Mara Alagic
Cambridge
IMECT 3
July
2002
Millionaire
Meredith, LuAnne & Dan
Adam the early bird
Starts at age 22
Invests $3000 for 6 years
At 11% interest
Mara Alagic
Cambridge
IMECT 3
July
2002
Millionaire
Meredith, LuAnne & Dan
Brenda the procrastinator
Starts investing at age 28
Invests $3000 every year thereafter
Until age 63
At 11% interest
Mara Alagic
Cambridge
IMECT 3
July
2002
Who retires with the most?
Meredith, LuAnne & Dan
Adam?
Brenda?
And how much difference between the
final results?
Plan A and B.xls
Mara Alagic
Cambridge
IMECT 3
July
2002
Meredith, LuAnne, & Dan
Mara Alagic
Cambridge
IMECT 3
July
2002
Invests $18,000
Retires at age 63 with
$1,128,249
Invests $108,000
Retires at age 63 with
$1,265,947
Meredith, LuAnne, & Dan
Mara Alagic
Cambridge
IMECT 3
July
2002
Graph of Sin  and Csc 
Sarah

Things to Consider
•
•
•

Differences?
•

Period?
Amplitude?
Asymptotes?
Between function
and its inverse
Green Globs
Mara Alagic
Cambridge
IMECT 3
July
2002
Sarah
Mara Alagic
Cambridge
IMECT 3
July
2002
Sarah
Cos and Sin Wave
Sine/Cosine Tracer
D. Bennett 3/91
D’
Cos
time
A
D
C
Sin
Geometer Sketchpad
Mara Alagic
Cambridge
IMECT 3
July
2002
Sarah
Tangent on a Circle
Tan
Tangent Tracer
D. Bennett 3/91
D
A
C
B
time
Geometer Sketchpad
Mara Alagic
Cambridge
IMECT 3
July
2002
Sarah
What if I still need help?
 Ask
for help
 Websites
•
•
http://www.drmath.com/
http://www.gomath.com/
 Play
Green Globs
AND…..
Mara Alagic
Cambridge
IMECT 3
July
2002
TO BECOME AN EXPERT OF
TRIG FUNCTIONS
Sarah
Mara Alagic
Cambridge
IMECT 3
July
2002