Transcript Agenda - RME in the Classroom

Length times width and Line times line
equals area
equals parabola
Incorporating two RME models into a cohesive
learning trajectory for quadratic functions
Fred Peck, University of Colorado and Boulder Valley School District
Jennifer Moeller, Boulder Valley School District
Agenda
• Realistic Mathematics Education
• A learning trajectory for quadratic functions
• Student work
• Extensions and open questions
“Mathematics should be thought of
as the human activity of
mathematizing
- not as a discipline of structures to
be transmitted, discovered, or even
constructed, but as schematizing,
structuring, and modeling the world
mathematically.”
Hans Freudenthal (as quoted in Fosnot & Jacob, 2010)
Five principles of RME
(Treffers, 1987)
• Mathematical exploration should take place within a
context that is recognizable to the student.
•
Models and tools should be used to bridge the
gap between informal problem-solving and formal
mathematics
• Students should create their own
and algorithms
procedures
• Learning should be social, and students should share
their solution processes, models, tools, and algorithms
with other students.
“Progressive formalization”
• Learning strands should be intertwined
Progressive formalization
• Students begin by mathematizing contextual
problems, and construct more formal
mathematics through guided re-invention
• Three broad levels:
– Informal: Models of learning: Representing mathematical
principles but lacking formal notation or structure (Gravemeijer, 1999)
– Preformal: Models for learning: Potentially generalizable
across many problems (Gravemeijer, 1999)
– Formal: Mathematical abstractions and abbreviations, often
far removed from contextual cues
The Iceberg Metaphor
preformal,
structured
formal notations
5+ 2= 7
(Webb, et al., 2008)
top of the iceberg
7
floating
capacity
5 2 3
5 2
informal,
experiential
©
F.M.- N.B.
The difficulty of applying RME principles to
quadratic functions
• In a word: context.
• We need a realistic context that students can
mathematize using informal reasoning, but
that can be re-invented into pre-formal
models and tools
• Why not projectile motion?
Two alternative contexts and models
1. Length times width equals area (Drijvers et al., 2010)
2. Line times line equals parabola (Kooij, 2000)
Formal
y
x
x
y
4
3
2
1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t
Pre-formal
h(t)
5
w
Informal
l
That’s an interesting graph…
http://viewpure.com/VSUKNxVXE4E
Length Width
0
1
2
3
4
5
6
7
8
9
10
10
9
8
7
6
5
4
3
2
1
0
Area
0
9
16
21
24
25
24
21
16
9
0
What patterns
do you see in
this table?
Input
(x)
Length
(l)
Input
(x)
Width
(w)
Input
(x)
Area
(A)
1
1
1
9
1
9
2
2
2
8
2
16
3
3
3
7
3
21
4
4
4
6
4
24
5
5
5
5
5
25
6
6
6
4
6
24
7
7
7
3
7
21
8
8
8
2
8
16
9
9
9
1
9
9
10
10
10
0
10
0
0
0
0
10
0
0
30
y
Line times Line equals Parabola
25
20
15
10
5
5
10
x
Is this
always true?
Explore what happens
when you multiply
two linear functions.
Do you always get a parabola?
What patterns do you notice?
The
vertex
of the
parabola
is halfway
between
the two
x-intercepts
The
concavity
of the
parabola
depends
on the slope
of the
two lines
The x-intercepts of the
parabola are the same as
those of the two lines
the What’s My Equation? game
There’s a parabola graphed on the next slide.
It’s your job to find the linear factors, and
then write the equation for the parabola.
Use your calculator to help!
What’s my equation?
y
8
7
6
5
4
3
2
1
–8
–7
–6
–5
–4
–3
–2
–1
1
–1
–2
–3
–4
–5
–6
–7
–8
2
3
4
5
6
7
8
x
What’s my equation?
y
11
10
9
8
7
6
5
4
3
2
1
–11
–10
–9
–8
–7
–6
–5
–4
–3
–2
–1
1
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
–11
2
3
4
5
6
7
8
9
10
11
x
What’s my equation?
y
11
10
9
8
7
6
5
4
3
2
1
–11
–10
–9
–8
–7
–6
–5
–4
–3
–2
–1
1
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
–11
2
3
4
5
6
7
8
9
10
11
x
Student work…
Formal
y
x
x
y
4
3
2
1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t
Pre-formal
h(t)
5
w
Informal
l
We use a JAVA applet from the
Freudenthal Institute to explore the
connections between
Line times line equals parabola
and
Length times width equals area
Use Google to search for “wisweb applets”
Select “Geometric algebra 2D”
Here, we can explore what line times line
equals parabola means in terms of our first
model: length times width equals area
Can you figure out how to construct an
area model for our last parabola:
From
standard form
to
factored form
Formal
y
x
x
y
4
3
2
1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t
Pre-formal
h(t)
5
w
Informal
l
Where do you see parabolas
in the real world?
How many parabolas do
you see in this movie?
http://viewpure.com/cnBf6HTizYc
The height (h) of the trampoline jumper at time t can be
modeled using the function:
Formal
y
x
x
y
4
3
2
1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t
Pre-formal
h(t)
5
w
Informal
l
Students have multiple
representations for quadratic
functions, and multiple methods to
convert between representations.
Formal
y
4
x
x
y
3
2
1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t
Pre-formal
h(t)
5
w
Informal
l
From graph to equation:
Line times line
equals parabola
Length times width
equals area
From equation
to graph:
Solving quadratic equations
Solving quadratic equations
In their own words…
Do the models that we’ve learned help
you solve problems?
Almost never
Sometimes
Often
0%
20%
40%
60%
80%
100%
In their own words…
Do the models that we’ve learned help
you understand formal mathematics?
Almost never
Sometimes
Often
0%
20%
40%
60%
80%
100%
Group discussion
• Extensions
Complete the square and vertex form
Polynomials
• Questions we have
Why is standard form compelling?
What are the downsides? How are
students impoverished?
References
Drijvers, P., Boon, P., Reeuwijk, M. van (2010). Algebra and Technology. In P. Drijvers
(ed.), Secondary Algebra Education: Revisiting Topics and Themes and Exploring
the Unknown. Rotterdam, NL: Sense Publishers. pp. 179-202
Fosnot, C. T., & Jacob, B. (2010). Young Mathematicians at Work: Constructing
Algebra. Portsmouth, NH: Heinemenn.
Gravemeijer, K. (1999). How emergent models may foster the constitution of formal
mathematics. Mathematical Thinking and Learning, 1(2), 155-177.
Kooij, H. van der (2000). What mathematics is left to be learned (and taught) with the
Graphing Calculator at hand? Presentation for Working Group for Action 11 at the
9th International Congress on Mathematics Education, Tokyo, Japan
Treffers, A. (1987). Three dimensions, a model of goal and theory description in
mathematics instruction-the Wiskobas Project. Dordrecht, The Netherlands: D.
Reidel.
Webb, D. C., Boswinkel, N., & Dekker, T. (2008). Beneath the Tip of the Iceberg: Using
Representations to Support Student Understanding. Mathematics Teaching in the
Middle School, 14(2), 4. National Council of Teachers of Mathematics.
Contact
Fred: [email protected]
Jen: [email protected]
Download the unit:
http://www.RMEInTheClassroom.com
Acknowledgements
We thank David Webb and Mary Pittman for introducing us to Realistic
Mathematics Education, and Henk van der Kooij and Peter Boon for
guiding us in the creation and implementation of this unit.